Vertex Cover Kernelization Revisited
Abstract
An important result in the study of polynomial-time preprocessing shows that there is an algorithm which given an instance (G,k) of Vertex Cover outputs an equivalent instance (G′,k′) in polynomial time with the guarantee that G′ has at most 2k′ vertices (and thus \(\mathcal{O}((k')^{2})\) edges) with k′≤k. Using the terminology of parameterized complexity we say that k-Vertex Cover has a kernel with 2k vertices. There is complexity-theoretic evidence that both 2k vertices and Θ(k^{2}) edges are optimal for the kernel size. In this paper we consider the Vertex Cover problem with a different parameter, the size \(\mathop{\mathrm{\mbox{\textsc{fvs}}}}(G)\) of a minimum feedback vertex set for G. This refined parameter is structurally smaller than the parameter k associated to the vertex covering number \(\mathop{\mathrm{\mbox {\textsc{vc}}}}(G)\) since \(\mathop{\mathrm{\mbox{\textsc{fvs}}}}(G)\leq\mathop{\mathrm{\mbox{\textsc{vc}}}}(G)\) and the difference can be arbitrarily large. We give a kernel for Vertex Cover with a number of vertices that is cubic in \(\mathop{\mathrm{\mbox{\textsc{fvs}}}}(G)\): an instance (G,X,k) of Vertex Cover, where X is a feedback vertex set for G, can be transformed in polynomial time into an equivalent instance (G′,X′,k′) such that |V(G′)|≤2k and \(|V(G')| \in\mathcal{O}(|X'|^{3})\). A similar result holds when the feedback vertex set X is not given along with the input. In sharp contrast we show that the Weighted Vertex Cover problem does not have a polynomial kernel when parameterized by the cardinality of a given vertex cover of the graph unless NP ⊆ coNP/poly and the polynomial hierarchy collapses to the third level.
Keywords
Kernelization Vertex cover Structural parameterization1 Introduction
A vertex cover of an undirected graph G is a subset of the vertices that contains at least one endpoint of every edge. An instance of the Vertex Cover problem consists of a graph G and integer k, and asks whether G has a vertex cover of size at most k. Vertex Cover is one of the six classic NP-complete problems discussed by Garey and Johnson in their famous work on intractability [26, GT1], and has played an important role in the development of parameterized algorithms [18, 19, 36]. A parameterized problem is a language L⊆Σ^{∗}×ℕ, and such a problem is (strongly uniform) fixed parameter tractable (FPT) if there is an algorithm to decide membership of an instance (x,k) in f(k)|x|^{c} time for some computable function f and constant c. Since Vertex Cover is such an elegant problem with a simple structure, it has proven to be an ideal testbed for new techniques in the context of parameterized complexity. The problem is also highly relevant from a practical point of view because of its role in bioinformatics [1] and other problem areas.
In this work we suggest a “refined parameterization” for the Vertex Cover problem using the feedback vertex number \(\mathop{\mathrm{\mbox{\textsc{fvs}}}}(G)\) as the parameter, i.e., the size of a smallest vertex set whose deletion turns G into a forest. We give a polynomial kernel for the unweighted version of Vertex Cover under this parameterization, and also supply a conditional superpolynomial lower bound on the kernel size for the variant of Vertex Cover where each vertex has a non-negative integral weight. But before we state our results we shall first survey the current state of the art for the parameterized analysis of Vertex Cover.
There has been an impressive series of ever-faster parameterized algorithms to solve k-Vertex Cover,^{1} which led to the current-best algorithm by Chen et al. that can decide whether a graph G has a vertex cover of size k in \(\mathcal{O}(1.2738^{k} +kn)\) time and polynomial space [9, 10, 20, 38]. Mishra et al. [34] studied the role of König deletion sets (vertex sets whose removal ensure that the size of a maximum matching in the remaining graph equals the vertex cover number of that graph) for the complexity of the Vertex Cover problem, and showed that Vertex Cover parameterized above the size of a maximum matching is fixed-parameter tractable by exhibiting a connection to Almost 2-SAT [40]. Gutin et al. [29] studied the parameterized complexity of various Vertex Cover-parameterizations above and below tight bounds which relate to the maximum degree of the graph and the matching size, obtaining FPT algorithms and hardness results. Raman et al. [39] gave improved algorithms for Vertex Cover parameterized above the size of a maximum matching: their algorithm decides in \(\mathcal{O}^{*}(9^{\ell})\) time whether a graph has a vertex cover of size m+ℓ, where m is the size of a maximum matching.
The Vertex Cover problem has also played an important role in the development of problem kernelization [28]. Kernelization is a concept that enables the formal mathematical analysis of data reduction through the framework of parameterized complexity. A kernelization algorithm (or kernel) is a polynomial-time procedure that reduces an instance (x,k) of a parameterized decision problem to an equivalent instance (x′,k′) such that |x′|,k′≤f(k) for some computable function f, which is the size of the kernel. We also use the term kernel to refer to the reduced instance (x′,k′).
The k-Vertex Cover problem admits a kernel with 2k vertices and \(\mathcal{O}(k^{2})\) edges, which can be obtained through crown reduction [2, 11, 12] or by applying a linear-programming theorem due to Nemhauser and Trotter [9, 35]. These kernelization algorithms have been a subject of repeated study and experimentation [1, 7, 16]. Very recently Soleimanfallah and Yeo [42] showed that for every constant c there exists a kernel with 2k−c vertices. This is mostly of theoretical interest however, since the running time of the kernelization algorithm is exponential in c.
There is some complexity-theoretic evidence that the size bounds for the kernel cannot be improved. Since all reduction rules found to date are approximation-preserving [36], it appears that a kernel with (2−ϵ)k vertices for any ϵ>0 would yield a polynomial-time approximation algorithm for Vertex Cover with a performance ratio 2−ϵ which would disprove the Unique Games Conjecture [32]. A breakthrough result by Dell and Van Melkebeek [15] shows that there is no polynomial kernel which can be encoded into \(\mathcal{O}(k^{2 - \epsilon})\) bits for any ϵ>0 unless NP ⊆ coNP/poly and the polynomial hierarchy collapses to the third level [44], which is reason to believe that the current bound of \(\mathcal{O}(k^{2})\) edges is tight up to k^{o(1)} factors.
This overview might suggest that there is little left to explore concerning kernelization for vertex cover, but this is far from true. The mentioned kernelization results use the requested size k of the vertex cover as the parameter. But there is no reason why we should not consider structurally smaller parameters, to see if we can preprocess instances of Vertex Cover such that their final size is bounded polynomially by such a smaller parameter, rather than by a function of the requested set size k. We study kernelization for the Vertex Cover problem using the feedback vertex number \(\mathop{\mathrm{\mbox {\textsc{fvs}}}}(G)\) as the parameter. Since every vertex cover is also a feedback vertex set we find that \(\mathop{\mathrm{\mbox{\textsc{fvs}}}}(G) \leq\mathop{\mathrm {\mbox{\textsc{vc}}}}(G)\) which shows that the feedback vertex number of a graph is a structurally smaller parameter than the vertex covering number: there are trees with arbitrarily large values of \(\mathop{\mathrm{\mbox{\textsc{vc}}}}(G)\) for which \(\mathop{\mathrm{\mbox{\textsc{fvs}}}}(G) = 0\). Observe that for difficult instances of Vertex Cover we have \(k \in\Theta(\mathop{\mathrm {\mbox{\textsc{vc}}}}(G))\) since the use of the 2-approximation algorithm immediately solves instances where \(k > 2\mathop{\mathrm{\mbox{\textsc{vc}}}}(G)\) or \(k < \mathop{\mathrm{\mbox{\textsc{vc}}}}(G) / 2\). Therefore we call our parameter “refined” since it is structurally smaller than the standard parameter for the Vertex Cover problem. Observe that the parameterization by \(\mathop{\mathrm{\mbox{\textsc{fvs}}}}(G)\) is not relevant for the setting of fixed-parameter algorithms, since it is dominated by various smaller parameters such as treewidth and the size of an odd cycle transversal, with respect to which Vertex Cover is still fixed-parameter tractable (see Sect. 5).
Our Results
Our contribution is twofold: we present a polynomial kernel, and a kernel lower bound for a structural parameterization of a weighted variant.
Upper Bounds
We prove that fvs-Vertex Cover has a kernel in which the number of vertices is bounded by min(2k,2|X|+28|X|^{2}+56|X|^{3}), which can be computed in \(\mathcal{O}(\sqrt{n} m + n^{5/3})\) time. The kernel size is at least as small as the current-best Vertex Cover kernel, but for graphs with small feedback vertex sets our bound can be expected to be significantly smaller.fvs-Vertex Cover
Instance: A simple undirected graph G, a feedback vertex set X⊆V(G) such that G−X is a forest, an integer k≥0.
Parameter: The size |X| of the feedback vertex set.
Question: Does G have a vertex cover of size at most k?
We also consider the problem fvs-Independent Set which is similarly defined: the difference is that we now ask whether G has an independent set of the requested size, instead of a vertex cover. Throughout this work k will always represent the total size of the set we are looking for; depending on the context this is either a vertex cover or an independent set. An instance (G,X,k) of fvs-Vertex Cover is equivalent to an instance (G,X,|V(G)|−k) of fvs-Independent Set which has the same parameter value and therefore the two problems are equivalent from a parameterized complexity and kernelization standpoint.
Lower Bounds
This lower bound parameterized by the cardinality of a given vertex cover is rather surprising, since Chlebík and Chlebíková used a modified form of crown reductions to prove that Weighted Vertex Cover parameterized by the target weight k admits a linear-vertex kernel [11]. In our construction for the lower bound we use only two different vertex weights: the value one, and a larger but polynomially-bounded value. Hence the comparative difficulty of the weighted problem does not stem from a tricky encoding of weights, but rather because the presence of weights allow us to encode complicated behavior (the OR of a series of inputs of Vertex Cover) into a graph which has a relatively simple structure (a small vertex cover). Section 5 contains a further discussion of kernelization for weighted problems. Observe that vc-Weighted Vertex Cover lies in FPT because the parameter is an upper bound on the treewidth of the input graph.vc-Weighted Vertex Cover
Instance: A simple undirected graph G, a weight function \(w: V(G) \to\mathbb{N^{+}}\), a vertex cover X⊆V(G), an integer k≥0.
Parameter: The cardinality |X| of the vertex cover.
Question: Is there a vertex cover C of G such that ∑_{v∈C}w(v)≤k?
Related Work
The idea of studying parameterized problems using alternative parameters is not new (see, e.g., [36]), but was recently advocated by Fellows et al. [22, 23, 37] in the call to investigate the complexity ecology of parameters. They posed that inputs to computational problems are rarely arbitrary or random because these inputs are created by processes which are themselves computationally bounded. This suggests that inputs might inherit structure from the processes which create them, possibly in unknown or unforeseen ways, and that we should therefore consider the complexity of problems not only when parameterized by their own solution value, but also by structural properties of the input, and in general by the optimum solution value to any other optimization problem on the instance. The main idea behind this program is therefore to determine how different parameters affect the parameterized complexity of a problem. Some recent results in this direction include FPT algorithms for graph layout problems parameterized by the vertex cover number of the graph [24] and an algorithm to decide isomorphism on graphs of bounded feedback vertex number [33]. There are a handful of applications of this idea to give polynomial kernels using alternative parameters. Fellows et al. [21, 23] show that the problems Independent Set, Dominating Set and Hamiltonian Circuit admit linear-vertex kernels on graphs G when parameterized by the maximum number of leaves in any spanning tree of G. A superset of the current authors [6] obtained a polynomial kernel for Treewidth parameterized by \(\mathop{\mathrm{\mbox {\textsc{fvs}}}}(G)\). Uhlmann and Weller [43] gave a polynomial kernel for Two-Layer Planarization parameterized by the feedback edge set number, which is a refined structural parameter for that problem since it is smaller than the natural parameter.
2 Preliminaries
In this work we only consider undirected, finite, simple graphs. For a graph G let V(G) be the vertex set and E(G) the edge set. We denote the independence number of G (i.e., the size of a maximum independent set) by α(G), the vertex covering number by \(\mathop{\mathrm{\mbox{\textsc{vc}}}} (G)\) and the feedback vertex number by \(\mathop{\mathrm{\mbox {\textsc{fvs}}}}(G)\). We will abbreviate maximum independent set as MIS, and feedback vertex set as FVS. For v∈V(G) we denote the open and closed neighborhoods of v by N_{G}(v) and N_{G}[v], respectively. For a set S⊆V(G) we have N_{G}(S):=⋃_{v∈S}N_{G}(v)∖S, and N_{G}[S]:=⋃_{v∈S}N_{G}[v]. The degree of a vertex v in graph G is denoted by deg_{G}(v). We write G′⊆G if G′ is a subgraph of G. For X⊆V(G) we denote by G[X] the subgraph of G that is induced by the vertices in X. The graph G[V(G)∖X] obtained from G by deleting the vertices in X and their incident edges is denoted by G−X.
A matching in a graph G is a set of edges M⊆E(G) such that no two distinct edges in M are incident on a common vertex. A matching is perfect if every vertex of the graph is incident on exactly one edge in the matching.
A vertex of degree one is called a leaf. If v is a vertex in a tree and v is not a leaf, then it is an internal node of the tree. The leaf set of a graph G is the set of degree-1 vertices, denoted by \(\mathop{\mathrm{\mbox{\textsc{Leaves}}}}(G):= \{v \in V(G) \mid\deg_{G}(v) = 1\}\). P_{2} is the graph consisting of a path on two vertices. We use [n] as a shorthand for {1,2,…,n}.
König’s Theorem
[41, Theorem 16.2]
For every bipartite graph G, the size of a minimum vertex cover equals the number of edges in a maximum matching.
Observation 1
- (i)
|V(F)|=2|M| andvc(F)=α(F)=|M|, since Fis bipartite.
- (ii)
Every vertex of Fis adjacent to at most one leaf.
- (iii)
If vis a leaf of F, then vhas a unique neighbor u∈V(F) and {u,v}∈M.
Observation 2
If G′ is a vertex-induced subgraph of graph Gthenα(G)≥α(G′).
Observation 3
If vis a leaf in Gthen there is a MIS for Gthat contains v.
3 Cubic Kernel for FVS-Vertex Cover
In this section we develop a cubic kernel for fvs-Vertex Cover. For the ease of presentation, we first develop a kernel for fvs-Independent Set. Using the correspondence between the two problems mentioned in the introduction, this kernel for fvs-Independent Set will immediately yield a kernel for fvs-Vertex Cover.
From now on we therefore focus on fvs-Independent Set. We first show that a single application of the Nemhauser-Trotter decomposition theorem [35], used for kernelization of the vertex cover problem by Chen et al. [9], allows us to restrict our attention to instances of fvs-Vertex Cover where the forest G−X has a perfect matching. This will greatly simplify the analysis of the kernel size as compared to the extended abstract of this work [30] where we worked with arbitrary forests G−X. In Sect. 3.1 we will then introduce a set of reduction rules and prove they are correct. Afterwards we will analyze the structure of the resulting reduced instances, in Sect. 3.2. This analysis will focus on conflict structures. An important ingredient in the kernel size bound will be a purely graph-theoretic extremal argument, which is developed in Sect. 3.3, and which will show that many conflict structures exist in reduced instances. As the last step we discuss the running time of a possible implementation of the reduction rules, and tie all ingredients together into a kernelization algorithm in Sect. 3.4.
So let us start by showing how to reduce to instances where the forest G−X has a perfect matching. For this purpose we re-state the Nemhauser-Trotter theorem here in terms of independent sets.
Proposition 1
[9, Proposition 2.1]
- 1.
if Iis a maximum independent set in G[V_{0}] then I∪Jis a maximum independent set in G, with J:=V(G)∖(C_{0}∪V_{0}), and
- 2.
α(G[V_{0}])≤|V_{0}|/2.
We will exploit the decomposition guaranteed by this proposition to show that after identifying a set of vertices which can be in any maximum independent set of G, there is a small (in terms of |X|) set I⊆V(G)∖X that we can add to X, such that the forest G−(X∪I) has a perfect matching.
Lemma 1
- 1.
G′−X′ has a perfect matching,
- 2.
|X′|≤2|X|, and
- 3.
k′≤k.
Proof
Given an instance (G,X,k) of fvs-Independent Set, use the algorithm of Proposition 1 to compute the two sets C_{0},V_{0}⊆V(G). Now set G′:=G[V_{0}], let \(\hat{X} := X \cap V_{0}\), and k′:=k−(|V(G)|−|V_{0}|−|C_{0}|). The proposition ensures that the instances (G,X,k) and \((G',\hat{X}, k')\) are equivalent, and it is easy to see that \(G' - \hat{X}\) is a forest since it is a subgraph of G−X. The last property of the proposition ensures that α(G′)≤|V(G′)|/2.
The fact that the forest G−X of an instance of fvs-Independent Set has a perfect matching is so useful that it warrants its own name.
Definition 1
An instance (G,X,k) of fvs-Independent Set is called clean if the forest G−X has a perfect matching.
We will apply Lemma 1 once at the start of our kernelization, and work on the resulting clean instance of the problem. The reduction rules we apply to shrink the instance further maintain the fact that the forest has a perfect matching.
3.1 Reduction Rules for Clean Instances
Consider a clean instance (G,X,k) of fvs-Independent Set, which asks whether a graph G with the FVS X has an independent set of size k. Throughout this section F:=G−X denotes the forest obtained by deleting the vertices in X, and recall that G−X has a perfect matching by the assumption that the instance is clean. To formulate our reduction rules we use the following notion.
Definition 2
(Chunks)
Let (G,X,k) be an instance of fvs-Independent Set. Define \(\mathcal{X}:= \bigl\{Y \subseteq X\bigm\vert\mbox{$Y$ is independent in~$G$ and~$0 < |Y| \leq2$}\bigr\}\) as the collection of chunks of X.
The chunks \(\mathcal{X}\) corresponding to an instance are size-≤2 subsets of the feedback vertex set X, which could be part of an independent set in G. Our first two reduction rules get rid of chunks when we can effectively determine that there is a MIS which does not contain them. We get rid of a chunk by either deleting it (when it is a single vertex) or by adding an edge (if a chunk consists of two non-adjacent vertices). Observe that after adding the edge {u,v} for u,v∈X the pair {u,v} is no longer independent, and therefore no longer counts as a chunk.
We rely on the fact that when given an independent subset X′⊆X of the feedback vertices, we can efficiently compute a largest independent set I in G which satisfies I∩X=X′: since such a set intersects X exactly in X′, and since it cannot use any neighbors of X′ the maximum size is |X′|+α(F−N_{G}(X′)) and this is polynomial-time computable since F−N_{G}(X′) is a forest. The following notion allows us to assess which chunks might occur in a MIS of G.
Definition 3
The number of conflicts \(\mathop{\mathrm{\mbox{\textsc {Conf}}}}_{F'}(X')\) induced by a subset X′⊆X on a subforest F′⊆F⊆G is defined as \(\mathop{\mathrm{\mbox{\textsc{Conf}}}}_{F'}(X'):= \alpha(F') - \alpha(F' - N_{G}(X'))\).
Reduction Rule 1
If there is a vertex v∈Xsuch that Conf_{F}({v})≥|X|, then delete vfrom the graph Gand from the set X.
Reduction Rule 2
If there are distinct vertices u,v∈Xwith \(\{u,v\} \not\in E(G)\)for whichConf_{F}({u,v})≥|X|, then add the edge {u,v} to G.
Since these two rules only affect the graph induced by X, they do not change the fact that forest F has a perfect matching. Correctness of the rules can be established from the following lemma.
Lemma 2
If X′⊆Xis a subset of feedback vertices such that Conf_{F}(X′)≥|X| then there is a MIS for Gthat does not contain all vertices of X′.
Proof
Assume that I⊆V(G) is an independent set containing all vertices of X′. We will prove that there is an independent set I′ which is disjoint from X′ with |I′|≥|I|. Since \(\mathop {\mathrm{\mbox{\textsc{Conf}}}}_{F}(X')\geq|X|\) it follows by definition that α(F)−α(F−N_{G}(X′))≥|X|; since I cannot contain any neighbors of vertices in X′ we know that |I∩V(F)|≤α(F−N_{G}(X′)), and since V(G)=X∪V(F) we have |I|≤|X|+α(F−N_{G}(X′))≤α(F). Hence the maximum independent set for F, which does not contain any vertices of X′, is at least as large as I; this proves that for every independent set containing X′ there is another independent set which is at least as large and avoids the vertices of X′. Therefore there is a MIS for G avoiding at least one vertex of X′. □
The next rule is used to remove trees from the forest F when the tree does not interact with any of the chunks in X.
Reduction Rule 3
If Fcontains a connected component T (which is a tree) such that for all chunks \(Y \in\mathcal{X}\)it holds that Conf_{T}(Y)=0, then delete Tfrom graph Gand decrease kbyα(T).
Since the rule deletes an entire tree from the forest F, it ensures that the remainder of the forest will have a perfect matching. To prove the correctness of Rule 3 we need the following lemma.
Lemma 3
Let Tbe a connected component of Fand let X_{I}⊆Xbe an independent set in G. IfConf_{T}(X_{I})>0 then there is a set X′⊆X_{I}with |X′|≤2 such that Conf_{T}(X′)>0.
Proof
Assume the conditions stated in the lemma hold. Recall that throughout this section we work on a clean instance, so let M be a perfect matching on T which exists since the forest F has a perfect matching. We will try to construct a MIS I for T that does not use any vertices in N_{G}(X_{I}); this must then also be a MIS for T−N_{G}(X_{I}) of the same size. By the assumption that \(\mathop{\mathrm {\mbox{\textsc{Conf}}}}_{T}(X_{I}) > 0\) any independent set in T must use at least one vertex in N_{G}(X_{I}) in order to be maximum, hence our construction procedure must fail somewhere; the place where it fails will provide us with a set X′ as required by the statement of the lemma.
Construction of a MIS
It is easy to see that a MIS of a tree with a perfect matching contains exactly one vertex from each matching edge. We now start building our independent set I for T that avoids vertices in N_{G}(X_{I}). To ensure I becomes a MIS for T, we need to add one endpoint of each edge in the matching M. If there is a vertex v in T such that N_{T}(v)={u} and N_{G}(v)∩X_{I}=∅, then the edge {v,u} must be in the matching M (since M is a perfect matching and there are no other edges incident on v). Because we must choose one of {u,v} in a MIS for T, and by Observation 3 choosing a degree-1 vertex will never conflict with choices that are made later on, we can add v to our independent set I while respecting the invariant that no vertex in I is adjacent in G to a vertex in X_{I}. Since we have then chosen one endpoint of the matching edge {u,v} in I, we can delete u,v and their incident edges to obtain a smaller graph T′ (which again contains a perfect submatching of M) in which we continue the process. As long as there is a vertex with degree one in T′ that has no neighbors in X_{I} then we take it into I, delete it and its neighbor, and continue. If this process ends with an empty graph, then by our starting observation the set I must be a MIS for T, and since it does not use any vertices adjacent to X_{I} it must also be a MIS for T−N_{G}(X_{I}); but this proves that α(T)=α(T−N_{G}(X_{I})) which means \(\mathop{\mathrm {\mbox{\textsc{Conf}}}}_{T}(X_{I}) = 0\), which is a contradiction to the assumption at the start of the proof. So the process must end with a non-empty graph T′⊆T such that vertices with degree one in T′ are adjacent in G to a vertex in X_{I} and for which the matching M restricted to T′ is a perfect matching on T′. We use this subgraph T′ to obtain a set X′ as desired.
Using the Subgraph to Prove the Claim
Consider a vertex v_{0} in T′ such that deg_{T′}(v_{0})=1, and construct a path P=(v_{0},v_{1},…,v_{2p+1}) by following edges of T′ that are alternatingly in and out of the matching M, until arriving at a degree-1 vertex whose only neighbor was already visited. Since T′ is acyclic, M restricted to T′ is a perfect matching on T′ and we start the process at a vertex of degree one, it is easy to verify that there is such a path P (there can be many; any arbitrary such path will suffice), that P contains an even number of vertices, that the first and last vertex on P have degree-1 in T′ and that the edges {v_{2i},v_{2i+1}} must be in M for all 0≤i≤p. Since we assumed that all degree-1 vertices in T′ are adjacent in G to X_{I}, there exist vertices x_{1},x_{2}∈X such that v_{0}∈N_{G}(x_{1}) and v_{2p+1}∈N_{G}(x_{2}). We now claim that X′:={x_{1},x_{2}} satisfies the requirements of the statement of the lemma, i.e., that \(\mathop{\mathrm{\mbox{\textsc {Conf}}}}_{T}(\{x_{1},x_{2}\}) > 0\). This fact is witnessed by considering the path P in the original tree T. Any MIS for T which avoids N_{G}({x_{1},x_{2}}) must use one endpoint of the matched edge {v_{0},v_{1}}, and since the choice of v_{0} is blocked because v_{0} is a neighbor to x_{1}, it must use v_{1}. But path P shows that v_{1} is adjacent in T to v_{2}, and hence we cannot choose v_{2} in the independent set. But since {v_{2},v_{3}} is again a matched edge, we must use one of its endpoints; hence we must use v_{3}. Repeating this argument shows that we must use vertex v_{2p+1} in a MIS for T if we cannot use v_{0}; but the use of v_{2p+1} is also not possible if we exclude N_{G}({x_{1},x_{2}}). Hence we cannot make a MIS for T without using vertices in N_{G}({x_{1},x_{2}}) which proves that α(T)>α(T−N_{G}({x_{1},x_{2}})). By the definition of conflicts this proves that \(\mathop{\mathrm {\mbox{\textsc{Conf}}}}_{T}(X') > 0\) for X′={x_{1},x_{2}}, which concludes the proof. □
Using this lemma we can prove the correctness of Rule 3. We remark that using a more involved argument based on a decomposition theorem describing independent sets in forests by Zito [45], it is possible to show that Lemma 3 holds even if F is a forest that does not admit a perfect matching. This argument can be found in an earlier version of this work [30, Lemma 4].
Lemma 4
Rule 3 is correct: if Tis a connected component in Fsuch that for all chunks \(Y \in\mathcal {X}\)it holds that Conf_{T}(X′)=0, thenα(G)=α(G−T)+α(T).
Proof
Assume the conditions in the statement of the lemma hold. It is trivial to see that α(G)≤α(G−T)+α(T). To establish the lemma we only need to prove that α(G)≥α(G−T)+α(T), which we will do by showing that any independent set I_{G−T} in G−T can be transformed to an independent set of size at least |I_{G−T}|+α(T) in G. So consider such an independent set I_{G−T}, and let X_{I}:=I_{G−T}∩X be the set of vertices which belong to both I_{G−T} and the feedback vertex set X. Suppose that α(T)>α(T−N_{G}(X_{I})). Then by Lemma 3 there is a subset Y⊆X_{I} with |Y|≤2 such that \(\mathop{\mathrm{\mbox{\textsc {Conf}}}}_{T}(Y) > 0\). Since X_{I} is an independent set, such a subset Y would also be independent, and hence would be a chunk in \(\mathcal{X}\). But by the preconditions to this lemma such a chunk Y does not exist and therefore we have α(T)=α(T−N_{G}(X_{I})).
Now we show how to transform I_{G−T} into an independent set for G of the requested size. Let I_{T} be a MIS in T−N_{G}(X_{I}), which has size α(T−N_{G}(X_{I}))=α(T). It is easy to verify that I_{G−T}∪I_{T} is an independent set in G because vertices of T are only adjacent to vertices of G−T which are contained in X. Hence the set I_{G−T}∪I_{T} is independent in G and it has size |I_{G−T}|+α(T). Since this argument applies to any independent set I_{G−T} in graph G−T it holds in particular for a MIS in G−T, which proves that α(G)≥α(G−T)+α(T). □
We introduce the concept of blockability for the statement of the last reduction rules.
Definition 4
The pair x,y∈V(G)∖X is X-blockable if there is a chunk \(Y \in\mathcal{X}\) such that {x,y}⊆N_{G}(Y).
This can be interpreted as follows: any independent set in G containing the chunk Y cannot contain x nor y, so using the chunk Y in an independent set blocks both vertices of the pair x,y from being in the same independent set. It follows directly from the definition that if x,y is not X-blockable, then for any combination of u∈N_{G}(x)∩X and v∈N_{G}(y)∩X we have u≠v and {u,v}∈E(G)—otherwise the singleton {u} would block x and y, or the pair {u,v} would be independent and would block x,y.
Reduction Rule 4
Delete vertices u,vwith their incident edges and decrease kby one.
If uhas a neighbor tin Fwhich is not v, make it adjacent to N_{G}(v)∩X.
If vhas a neighbor win Fwhich is not u, make it adjacent to N_{G}(u)∩X.
If the vertices t,wexist then they are unique; add the edge {t,w} to the graph.
It is not hard to see that this rule does not change the fact that F has a perfect matching: if the edge {u,v} was contained in the perfect matching, then the matching restricted to the remaining vertices is a perfect matching for the remaining graph. If {u,v} was not contained in the perfect matching then u was matched to t and v was matched to w; we obtain a perfect matching for the reduced graph by matching t to w, using the edge that is added to the graph by the reduction rule.
Lemma 5
Let (G,X,k) withF:=G−Xbe an instance to which Rule 4 is applicable at vertices u,v, and let (G′,X,k−1) be the instance resulting from the reduction. Then it holds thatα(G)≥k⇔α(G′)≥k−1.
Proof
Assume the conditions in the statement of the lemma hold. We prove the two directions separately.
(⇒) Let I_{G} be an independent set for graph G of size at least k. We show how to obtain an independent set I_{G′} for graph G′ of size at least |I_{G}|−1≥k−1. Observe that no independent set in G can contain both {u,v} since they are adjacent. If I_{G} does not contain any of the vertices {u,v} then we show how to obtain \(I'_{G}\) which is at least as large and does contain one of {u,v}; so assume I_{G} avoids u and v. Since the pair u,v is not X-blockable by the preconditions for the reduction rule, we know that there is at least one vertex among u,v for which no neighbor in X is chosen in I_{G}. Assume without loss of generality (by symmetry) that this holds for u, such that N_{G}(u)∩X∩I_{G}=∅. Since v is not in I_{G} by assumption, the only neighbor of u that can be in I_{G} is its neighbor t in F unequal to v (if such a t exists; see Fig. 2). If no such t exists then \(I'_{G} := I_{G} \cup\{u\}\) is a bigger independent set in G; otherwise \(I'_{G} := (I_{G} \setminus\{t\}) \cup\{u\}\) is an equally large independent set. So using this replacement argument and symmetry, we may assume that I_{G} is an independent set of size at least k for G that contains u but not v.
We now claim that I_{G′}:=I_{G}∖{u} is an independent set of size ≥k−1 in G′. Since it is easy to see that I_{G′} has the desired size, it remains to show that it is an independent set in G′. To establish this we need to show that the transformation to G′ does not add any edges between vertices of I_{G′}. This is ensured because all edges that are added by the transformation have at least one endpoint which is a neighbor of u: all added edges are either incident on w or a vertex in N_{G}(u)∩X. Hence for each added edge one endpoint z is adjacent to u, and since we assumed u∈I_{G} this implies that z cannot be in I_{G′} since I_{G′} is a subset of the independent set I_{G} in G and having adjacent vertices u and z in I_{G} would violate independence. Therefore I_{G′} is indeed an independent set of the required size in G′.
If vertex t exists and t∈I_{G′}, then define I_{G}:=I_{G′}∪{v}. To prove I_{G} is an independent set in G we show that N_{G}(v)∩I_{G′}=∅ by consecutively proving that {u,w}∩I_{G′}=∅ and N_{G}(v)∩X∩I_{G′}=∅, which together suffice to establish our claim because N_{G}(v)={u,w}∪(N_{G}(v)∩X) (for as far as t exists). Since \(u \not\in V(G')\) we trivially have \(u \not\in I_{G'}\), and because the edge {t,w} is added when forming G′ and t∈I_{G′} by the case distinction we have \(w \not\in I_{G'}\). To see that N_{G}(v)∩X∩I_{G′}=∅ observe that N_{G}(v)∩X⊆N_{G′}(t) by the construction of G′, and since t∈I_{G′} and I_{G′} is independent in G′ this proves the claim and the correctness of this case.
If vertex w exists and w∈I_{G′}, then define I_{G}:=I_{G′}∪{u}. The correctness argument is symmetric to that of the previous case.
In the remaining case we know that {t,w}∩I_{G′}=∅. There must be some z∈{u,v} such that N_{G}(z)∩X∩I_{G′}=∅; because if there is no such z then by combining one vertex from N_{G}(u)∩X∩I_{G′} and one from N_{G}(v)∩X∩I_{G′} gives a pair which proves that {u,v} is X-blockable in G, contradicting the precondition to the reduction rule. We now assign I_{G}:=I_{G′}∪{z}. Since N_{G}(z)∩F⊆{t,u,v,w} and these vertices either do not exist in G′ or are not in I_{G′} by the case distinction, we know {t,u,v,w}∩I_{G′}=∅. Since N_{G}(z)∩X∩I_{G′}=∅ by our choice of z this proves that the addition of z to the independent set does not violate independence, because N_{G}(z)⊆(N_{G}(z)∩X)∪{t,u,v,w}.
Reduction Rule 5
Delete {t,u,v,w} and their incident edges from Gand decrease kby two.
Make padjacent to all vertices of N_{G}(t)∩X.
Make qadjacent to all vertices of N_{G}(w)∩X.
Once again it is not difficult to see that the rule preserves the fact that F has a perfect matching: since t and w have degree one in F, they must be matched to u and v in a perfect matching; hence the rule effectively deletes the endpoints of two matching edges from the graph.
Lemma 6
Let (G,X,k) withF:=G−Xbe an instance to which Rule 5 is applicable at vertices t,u,v,w, and let (G′,X,k−2) be the instance resulting from the reduction. Then it holds thatα(G)≥k⇔α(G′)≥k−2.
Proof
Assume the conditions in the statement of the lemma hold. We prove the two directions separately.
If I_{G}∩X∩N_{G}(t)=∅ and I_{G}∩X∩N_{G}(w)=∅ then define \(I'_{G} := (I_{G} \setminus\{u,v,t,w\})\cup\{t,w\}\) which is easily seen to be an independent set. Since no independent set can contain three or more vertices from {u,v,t,w} (because of the edges {u,t} and {v,w}) we now have \(|I'_{G}| \geq|I_{G}|\).
If I_{G}∩X∩N_{G}(t)≠∅ then we must have I_{G}∩X∩N_{G}(w)=∅; for if both sets are non-empty, then taking one vertex from I_{G}∩X∩N_{G}(t) and one vertex from I_{G}∩X∩N_{G}(w) yields a pair which shows that {t,w} is X-blockable, which contradicts the preconditions to Rule 5. Using the same argument we must have that I_{G}∩X∩N_{G}(u)=∅, otherwise {t,u} is X-blockable. Set \(I'_{G} := (I_{G} \setminus\{p,u,t,v,w\}) \cup\{u,w\}\). The neighborhood conditions show that no neighbors of u,w in X are contained in I_{G} (and hence in I_{G′}), and because we explicitly delete any neighbors that u,w might have in F when forming \(I'_{G}\) we see that \(I'_{G}\) is also an independent set in G. If I_{G}∩X∩N_{G}(t)≠∅ as specified by the precondition for this case, then we cannot have t∈I_{G} because then I_{G} would not be independent. The edges {p,u} and {v,w} in G show that of the set {p,u,v,w} at most two vertices are in an independent set; hence in this situation I_{G} contains at most two vertices from {p,u,t,v,w} and therefore we have \(|I'_{G}| \geq|I_{G}|\).
If I_{G}∩X∩N_{G}(w)≠∅ then we must have that I_{G}∩X∩N_{G}(t)=I_{G}∩X∩N_{G}(v)=∅, and we set \(I'_{G} := (I_{G} \setminus\{q,u,t,v,w\}) \cup\{t,v\}\). The correctness argument is symmetric to that of the previous case.
If t,w∈I_{G} then define I_{G′}:=I_{G}∖{t,w}. Since t,w∈I_{G} implies that \(u,v \not\in I_{G}\) we know that all vertices in I_{G′} still exist in G′. It remains to show that they form an independent set there. Because the reduction to G′ only adds edges incident on p and q, it suffices to show that for all edges incident on p or q which are added by the reduction there is at least one endpoint not in I_{G′}. The transformation from G to G′ adds edges from N_{G}(t)∩X to p, and edges from N_{G}(w)∩X to q. But since t,w∈I_{G} we know that the independent set I_{G} contains no vertices of N_{G}(t)∩X or N_{G}(w)∩X, and hence the defined set I_{G′} is an independent set in G′.
If t,v∈I_{G} then define I_{G′}:=I_{G}∖{t,v}. All vertices in I_{G′} must exist in G′ since u,w cannot be in I_{G} because their neighbors t,v are in I_{G}. The edges we add in the transformation to G′ do not violate independence: because t∈I_{G} we have N_{G}(t)∩I_{G}=∅, and similarly because v∈I_{G} we have N_{G}(v)∩I_{G}=∅ which in particular means \(q \not\in I_{G}\). For all edges that we add, at least one endpoint is not in I_{G} and therefore not in I_{G′}; this proves that I_{G′} is an independent set in G′.
If u,w∈I_{G} then define I_{G′}:=I_{G}∖{u,w}. The proof of correctness is symmetric to that for the previous case.
If N_{G}(t)∩X∩I_{G′}=∅ and N_{G}(w)∩X∩I_{G′}=∅ then assign I_{G}:=I_{G′}∪{t,w}. Since vertices t,w are clearly non-adjacent in G, and because the vertices in I_{G′} form an independent set in G (as the transformation to G does not add edges between vertices in I_{G′}) we now have that I_{G} is an independent set in G of the required size.
If N_{G}(t)∩X∩I_{G′}≠∅ then we must have N_{G}(w)∩X∩I_{G′}=∅, otherwise taking one vertex from N_{G}(t)∩X∩I_{G′} and one from N_{G}(w)∩X∩I_{G′} would give a pair which shows that {t,w} is X-blockable in the original graph G, which contradicts the preconditions for Rule 5. Similarly we must have N_{G}(u)∩X∩I_{G′}=∅ by the assumption that {u,t} is not X-blockable in G. Since vertex p is adjacent in G′ to all vertices of N_{G}(t)∩X, we know that by independence of I_{G′} if N_{G}(t)∩X∩I_{G′}≠∅ then \(p \not\in I_{G'}\). We now set I_{G}:=I_{G′}∪{u,w} which must form an independent set in G because the established conditions show that none of the vertices of N_{G}({u,w}) can be in I_{G′}. It is easy to see that |I_{G}|≥k in this case.
If N_{G}(w)∩X∩I_{G′}≠∅ then we must have N_{G}(t)∩X∩I_{G′}=N_{G}(v)∩X∩I_{G′}=∅ by the non-blockability of {w,t} and {w,v}. We assign I_{G}:=I_{G′}∪{t,v}. The correctness proof is symmetric to that of the previous case.
3.2 Structure of Reduced Instances
When no reduction rules can be applied to an instance, we call it reduced. The main purpose of this section is to prove that in reduced clean instances, the number of vertices in the forest F is at most cubic in the size of the feedback vertex set. We sketch the main idea behind this analysis.
The analysis is based on the idea of identifying conflict structures in the forest G−X. Informally, one may think of a conflict structure S as a subgraph of the forest F which bears witness to the fact that there is a chunk \(Y \in\mathcal {X}\) such that an independent set in G which contains Y, contains less vertices from S than an optimal independent set in F. Hence this conflict structure shows that by choosing Y to be a part of an independent set, we pay for it inside the conflict structure S. Since we trigger a reduction rule once there is a chunk \(Y \in\mathcal {X}\) which induces at least |X| conflicts (i.e., for which we have to pay at least |X|), there cannot be too many conflict structures in a reduced instance. The following notion is important to make these statements precise.
Definition 5
Define the number of active conflicts induced on the forest F by the chunks \(\mathcal{X}\) as \(\mathop{\mathrm {\mbox{\textsc{Active}}}}_{F}(\mathcal {X}) := \sum_{Y \in\mathcal{X}}\mathop{\mathrm{\mbox{\textsc{Conf}}}}_{F}(Y)\).
So the number of active conflicts is simply the number of conflicts induced on F summed over all chunks of the instance. For reduced instances, this value is cubic in |X|.
Observation 4
Let (G,X,k) be a reduced instance. By Rule 1 every v∈XsatisfiesConf_{F}({v})<|X|, and by Rule 2 every pair of distinct non-adjacent vertices {u,v}⊆XsatisfiesConf_{F}({u,v})<|X|. Hence \(\textsc{Active}_{F}(\mathcal{X}) \leq|X|^{2} + \binom{|X|}{2}|X|\).
The global argument to bound the kernel size is therefore to show that in a reduced instance with forest F, the number of conflict structures that can be found is linear in the size of the forest. Since the total number of conflicts that are induced by chunks \(\mathcal{X}\) (the number of active conflicts) is bounded by \(\mathcal {O}(|X|^{3})\), this will prove that the number of vertices in F is \(\mathcal{O}(|X|^{3})\).
The proof of the kernel size bound is organized as follows. In the remainder of this section we will formally define conflict structures, and prove that the number of active conflicts induced on the forest F grows linearly with the number of conflict structures contained in F. We give an extremal graph-theoretic result showing that any forest with a perfect matching contains linearly many conflict structures, in Sect. 3.3. As the final step we will combine these results with Observation 4 to give the kernel size bound in Sect. 3.4.
Definition 6
(Conflict Structures)
A conflict structure of type A in F is a pair of distinct vertices {v_{1},v_{2}} such that {v_{1},v_{2}}∈M and deg_{F}(v_{1}),deg_{F}(v_{2})≤2.
A conflict structure of type B in F is a path on four vertices (v_{1},v_{2},v_{3},v_{4}) such that v_{1} and v_{4} are leaves of F, and deg_{F}(v_{2})=deg_{F}(v_{3})=3.
Observe that in a conflict structure of type B, the edges {v_{1},v_{2}} and {v_{3},v_{4}} must be contained in the perfect matching M by Observation 1. Although conflict structures can be defined for arbitrary forests with a perfect matching, we are of course interested in the forests that occur in a reduced clean instance of fvs-Independent Set. To capture the interaction between chunks of such an instance and conflict structures in the forest, we need the following definition.
Definition 7
(Hitting conflict structures)
\(Y \in\mathcal{X}\)hits a conflict structure {v_{1},v_{2}} of type A in F if {v_{1},v_{2}}⊆N_{G}(Y).
- \(Y \in\mathcal{X}\)hits a conflict structure (v_{1},v_{2},v_{3},v_{4}) of type B in F if one of the following holds:
{v_{1},v_{2}}⊆N_{G}(Y), or
{v_{3},v_{4}}⊆N_{G}(Y), or
{v_{1},v_{4}}⊆N_{G}(Y).
The importance of Reduction Rules 4 and 5 now becomes clear.
Observation 5
If (G,X,k) is a reduced clean instance offvs-Independent Setand Sis a conflict structure in a tree Tof the forest F:=G−X, then Sis hit by some chunk of \(\mathcal{X}\): if a structure of type Ais not hit this triggers Rule 4, and if a structure of type Bis not hit this triggers Rule 5.
The fact that each conflict structure is hit by at least one chunk in a reduced instance, allows us to relate the number of vertex-disjoint conflict structures to the number of active conflicts that must be induced by the chunks.
Lemma 7
Let (G,X,k) be a reduced clean instance offvs-Independent Setwith forest F:=G−Xsuch that Mis a perfect matching in F, and let \(\mathcal{S}\)be a set of vertex-disjoint conflict structures in F. Then\(\textsc{Active}_{F}(\mathcal{X}) \geq|\mathcal{S}|\).
Proof
Assume the conditions in the statement of the lemma hold. Consider some chunk \(Y \in\mathcal{X}\), and let \(\mathcal {S}_{Y}\) be the structures in \(\mathcal{S}\) which are hit by Y according to Definition 7. We will first show that \(\mathop{\mathrm{\mbox{\textsc{Conf}}}}_{F}(Y) \geq|\mathcal{S}_{Y}|\), and later we will show how this implies the lemma.
- 1.
If S={v_{1},v_{2}} is a conflict structure of type A, then {v_{1},v_{2}}⊆N_{G}(Y) by Definition 7 since Y hits S, and edge {v_{1},v_{2}} is contained in M by Definition 6. Now obtain F′ from F by deleting the vertices v_{1} and v_{2}, and obtain M′ from M by deleting the edge {v_{1},v_{2}}.
- 2.If S=(v_{1},v_{2},v_{3},v_{4}) is a conflict structure of type B, then the edges {v_{1},v_{2}} and {v_{3},v_{4}} are contained in M by Observation 1. By Definition 7, using the fact that Y hits S, one of the following applies:Let F′ be the resulting graph, and M′ the resulting matching.
If {v_{1},v_{2}}∈N_{G}(Y) then delete vertices v_{1},v_{2} from F and delete the edge between them from M.
If {v_{3},v_{4}}∈N_{G}(Y) then delete vertices v_{3},v_{4} from F and delete the edge between them from M.
If {v_{1},v_{4}}∈N_{G}(Y) then delete vertices v_{1},v_{4} from F, delete the edges {v_{1},v_{2}} and {v_{3},v_{4}} from M and replace them by the edge {v_{2},v_{3}}.
We have shown that for every chunk \(Y \in\mathcal {X}\) it holds that \(\mathop{\mathrm{\mbox{\textsc {Conf}}}}_{F}(Y) \geq|\mathcal{S}_{Y}|\), where \(\mathcal{S}_{Y}\) is the set of conflict structures hit by Y. The lemma now follows from the definition of active conflicts as the sum of the conflict values over all chunks, using that all conflict structures in \(\mathcal{S}\) are hit by at least one chunk (Observation 5). This concludes the proof. □
The previous lemma shows that if F has many conflict structures, then the number of active conflicts must be large, and therefore the size of the feedback vertex set must be large. The extremal argument of the next section makes it possible to turn this relation into a kernel size bound.
3.3 Packing Conflict Structures
In this section we present an extremal result which shows that trees with a perfect matching contain linearly many conflict structures.
Theorem 1
Let Tbe a tree with a perfect matching. There is a set\(\mathcal {S}\)of mutually vertex-disjoint conflict structures in Twith \(|\mathcal{S}| \geq|V(T)| / 14\).
Proof
If T is the tree on two vertices then the statement follows trivially, since T contains exactly one conflict structure of type A (see Definition 6). In the remainder we therefore assume that T≠K_{2} which implies that T has at least four vertices: the number of vertices must be even, since T has a perfect matching. We use a proof by construction which finds a set of conflict structures. The procedure grows a subtree T′⊆T and set \(\mathcal{S}\) incrementally, and during each augmentation step of the tree we enforce an incremental inequality which shows that the number of vertices of T which are contained in T′, is proportional to the number of conflict structures found so far in the subtree T′. This proof strategy is inspired by the method of “amortized analysis by keeping track of dead leaves” which is used in extremal graph theory [27].
So the proof revolves around a subtree T′⊆T that is grown by successively adding vertices to it. We use the following characteristics of the subgraph T′ in the analysis. The vertices \(\mathop{\mathrm{\mbox{\textsc{Leaves}}}}(T') \setminus\mathop {\mathrm{\mbox{\textsc{Leaves}}}}(T)\) are the open branches of T′. The open branches are essentially the vertices on the boundary of the subgraph T′, where we will eventually “grow” the subtree T′ to make it larger, until it encompasses all of T. Observe that when we have grown the tree T′ until it equals T, then the number of open branches is 0 by definition. We use the letter O to denote the number of open branches of the current state of the subtree T′. While growing the subtree we construct a set \(\mathcal {S}\) of vertex-disjoint conflict structures. We use C as an abbreviation for \(|\mathcal{S}|\). It turns out that certain vertices of the tree T play a special role in the amortized analysis that is implicit in the proof. We call these vertices spikes.
Definition 8
A spike in tree T is a vertex v such that deg_{T}(v)=3 and there is exactly one leaf of T adjacent to v. A vertex v∈V(T) is a live spike with respect to the current subtree T′ if v is a spike in T and an open branch of T′.
- (i)
For all conflict structures \(S \in\mathcal{S}\) it holds that \(V(S) \subseteq V(T') \setminus(\mathop{\mathrm{\mbox{\textsc {Leaves}}}}(T') \setminus\mathop{\mathrm{\mbox{\textsc{Leaves}}}}(T))\), i.e., the vertices we use in conflict structures are contained in T′ and are not open branches of T′.
- (ii)
All vertices of T′ which have a neighbor outside T′ are leaves of T′, implying that when |V(T′)|≥2 all vertices of T′ which have a neighbor outside T′ are open branches of T′.
The first part of the invariant will ensure that the conflict structures we find are mutually vertex-disjoint. The second part of the invariant is important because it implies that if T′ has no open branch vertices, then T′ coincides with T. It is trivial to see that the invariants are initially satisfied for an empty tree T′ and empty set of conflict structures \(\mathcal{S}\). We will now describe the augmentation operations. Whenever we talk about the neighbors of a vertex v in this description, we mean v’s neighbors in the graph T unless explicitly stated otherwise. Similarly, when we talk about a vertex being a leaf then we mean a leaf of the tree T, rather than T′.
Initialization
The first operation we describe shows how to initialize the subtree T′. Recall from the beginning of the proof that we could assume |V(T)|≥4.
Operation 1
Let vbe a leaf of Tand let ube its neighbor in the tree. Initialize T′ as the tree on vertex set N_{T}[u].
Claim 1
Operation 1 satisfies the incremental inequality and maintains the invariants.
Proof
For an empty tree we obviously have O=S=C=N=0. Let us now consider how these values are affected by the tree initialization. Since T is connected and has at least four vertices, u has at least one neighbor other than v. We claim that all vertices N_{T}(u)∖{v} are open branches of T′ after the initialization. By Observation 1 vertex v is the only leaf adjacent to u, and since T is a tree, the subtree induced by vertex set N_{T}[u] has the vertices N_{T}(u) as leaves. Therefore the vertices N_{T}(u)∖{v} are contained in \(\mathop{\mathrm {\mbox{\textsc{Leaves}}}}(T')\setminus\mathop{\mathrm{\mbox{\textsc{Leaves}}}}(T)\) and are open branches of T′ by definition, so ΔO=|N_{T}(u)−1|. The number of vertices added to the tree by the initialization is exactly ΔN=|N_{T}[u]|. The number of live spikes cannot decrease by this operation (since it started at zero, and cannot become negative); hence ΔS≥0. Since we do not add any conflict structures to \(\mathcal{S}\) we find ΔC=0. It is easy to see that this combination of values satisfies the incremental inequality since |N_{T}(u)−1|≥1. Since we do not add conflict structures, invariant (3.3) is trivially maintained. Invariant (3.3) is maintained by adding all neighbors of u to the tree simultaneously. □
Observe that the initialization ensures that tree T′ has at least three vertices, which will be used later on.
Augmentation
We will now describe the operations which are used to augment the tree once it is initialized. For each augmentation we prove that it satisfies the incremental inequality. After describing the remaining four operations, we prove that whenever the tree T′ does not yet encompass all of T, then some augmentation is applicable. When describing the augmentation steps of the subtree T′ we will use \(T'_{a}\) to refer to the status of the tree before the augmentation, and \(T'_{b}\) to refer to its status after the augmentation. When the intended meaning is clear from the context we will just write T′.
Operation 2
If |V(T′)|≥3 and there is a vertex v_{0}∈V(T′) with deg_{T}(v_{0})=2 such that N_{T}(v)∖V(T′) contains a spike vertex v_{1}, then add v_{1}to T′.
Claim 2
Operation 2 satisfies the incremental inequality and maintains the invariants.
Proof
The number of vertices in T′ increases by exactly one. Since deg_{T}(v_{0})=2, the vertex v_{0} is not a spike. Therefore the number of live spikes increases by one through this operation (ΔS=1) since the spike v_{1} becomes an open branch by this augmentation: v_{1} will be a leaf of T′, yet is not a leaf of T since deg_{T}(v_{1})=3 by definition of a spike. The number of vertices increases by one (ΔN=1). The number of open branches does not change: vertex v_{0} is lost as an open branch, but instead v_{1} becomes an open branch (ΔO=0). Since the number of conflict structures does not change (ΔC=0) it is now trivial to see that these values satisfy the inequality. Since we do not add conflict structures we maintain invariant (3.3). Invariant (3.3) is maintained because prior to the augmentation, vertex v_{1} is the only neighbor of v_{0} which is not yet contained in T′ which follows from the fact that v_{0} must have a parent in the tree T′ because |V(T′)|≥3, and the degree of v_{0} is only two. So the augmentation effectively adds all vertices N_{T}[v_{0}] to T′. □
The remaining augmentation operations grow the subtree by extending it over a path.
Definition 9
A tree extending path is a path P=(v_{0},v_{1},…,v_{q}) in T such that V(P)∩V(T′)={v_{0}} and v_{0} is an open branch vertex of T′.
Operation 3
If |V(T′)|≥3 and there is a tree extending path P=(v_{0},v_{1}) such that v_{0}and v_{1}are adjacent to leaves l_{0},l_{1}of Trespectively with \(l_{0}, l_{1} \not\in V(T')\)and deg_{T}(v_{0})=deg_{T}(v_{1})=3, then add the vertices N_{T}[V(P)] to the tree T′, and add the conflict structure of type Bcontaining (l_{0},v_{0},v_{1},l_{1}) to \(\mathcal{S}\).
Claim 3
Operation 3 satisfies the incremental inequality and maintains the invariants. The added conflict structure is disjoint from previously found structures.
Proof
Before the operation, vertex v_{0} is already contained in T′ and has a unique neighbor p inside T′ since v_{0} is a leaf of the tree T′ which has at least two vertices. Observe that p cannot be a leaf of T′, since v_{0} is a leaf of T′ and |V(T′)|≥3. Hence the neighbors of v_{0} in T are exactly {p,l_{0},v_{0}}. Similarly, the neighbors of v_{1} in T are exactly {q,l_{1},v_{0}} for a vertex \(q \not\in V(T')\) which is not a leaf of T. Therefore the vertices which are added to T′ by this operation, and which were not contained in T′ already, are exactly {l_{0},l_{1},v_{1},q} which shows that ΔN=4. Now consider the effect of the augmentation on the number of live spike vertices. Vertex v_{0} is a live spike in T′ before the augmentation: it is an open branch vertex by definition of a tree extending path, and the degree and leaf requirements of Definition 8 are met. Vertex v_{0} becomes an internal vertex of T′ by adding its neighbors to the tree, and therefore it will no longer be a live spike after the augmentation. But no other live spikes can be lost by the augmentation, hence ΔS≥−1. Since we add a conflict structure in this operation, ΔC=1. Let us finally consider the effect of this operation on the number of open branches. Clearly vertex v_{0} is no longer an open branch after the augmentation, and it was one before the augmentation. Vertices l_{0} and l_{1} are leaves of T and therefore do not become open branch vertices. But the vertex q cannot be a leaf of T by Observation 1, and it will be a leaf of T′ after the augmentation. Hence the loss of v_{0} as an open branch is compensated by q becoming an open branch, and ΔO=0. It is trivial to see that this combination of values satisfies the incremental inequality.
Invariant (3.3) is maintained by adding the closed neighborhood of a path to the tree T′, ensuring that afterwards no vertex on the path P can have neighbors outside T′. Adding N_{T}[V(P)] to T′ ensures that after the augmentation, none of the vertices of (l_{0},v_{0},v_{1},l_{1}) can be open branches of T′ while all those vertices are contained in T′, which shows how invariant (3.3) is maintained. By the same invariant, none of the vertices {l_{0},v_{0},v_{1},l_{1}} are contained in conflict structures in \(\mathcal{S}\) prior to the augmentation, since the involved vertices are not in T′ or open branches of T′. Hence the structure we add does not intersect any other structures in the set. □
Operation 4
If |V(T′)|≥3 and there is a tree extending path P=(v_{0},…,v_{q}) for q≤2 such that deg_{T}(v_{q−1}),deg_{T}(v_{q})≤2, and the edge between v_{q−1}and v_{q}is contained in the perfect matching in T, then add the vertices N_{T}[V(P)] to the tree T′, and add the conflict structure {v_{q−1},v_{q}} to \(\mathcal{S}\).
Claim 4
Operation 4 satisfies the incremental inequality and maintains the invariants. The added conflict structure is disjoint from previously found structures.
Proof
Operation 5
If |V(T′)|≥3 and there is a tree extending path P=(v_{0},…,v_{q}) for q≤2 such that (a) deg_{T}(v_{q})≥4 or (b) deg_{T}(v_{q})=3 and v_{q}is not adjacent to a leaf of T, then add the vertices N_{T}[V(P)] to the tree T′.
Claim 5
Operation 5 satisfies the incremental inequality.
Proof
- 1.
If (a) deg_{T}(v_{0})≥4 or (b) deg_{T}(v_{0})=3 and v_{0} is not adjacent to a leaf of T, then Operation 5 is applicable to the tree extending path (v_{0}).
- 2.If deg_{T}(v_{0})=3 and v_{0} is adjacent to a leaf of T, then consider some neighbor v_{1}∈N_{T}(v_{0})∖V(T′) which is not a leaf of T. Since v_{0} has exactly one neighbor in T′ and is adjacent to exactly one leaf of T (by Observation 1), such a vertex exists. Now consider the maximal path P=(v_{0},v_{1},…,v_{q}) obtained by starting with the edge {v_{0},v_{1}} and following vertices which have degree two in T, until arriving at the first vertex v_{q} which has deg_{T}(v_{q})≠2. If deg_{T}(v_{1})≠2 then this simply results in P=(v_{0},v_{1}). Observe that by this definition, vertices v_{1},…,v_{q} are not contained in T′.
- (a)
If (a) deg_{T}(v_{1})≥4 or (b) deg_{T}(v_{1})=3 and v_{1} is not adjacent to a leaf of T, we find that Operation 5 is applicable to the tree extending path (v_{0},v_{1}).
- (b)
If v_{1} has degree three, then by the previous case it is adjacent to a leaf in T. Operation 3 is applicable to the tree extending path (v_{0},v_{1}). Observe that the leaf of T adjacent to v_{0} cannot be contained in T′, as per the discussion above.
- (c)Since v_{1} has degree at least two in T by our choice of v_{1} as not being a leaf of T, in the remaining situations we have deg_{T}(v_{1})=2 and therefore there exists some v_{2} on the path P we defined earlier. Now observe that T having a perfect matching implies that v_{2} cannot be adjacent to a leaf of T: by definition of this case, v_{0} is adjacent to a leaf of T. If v_{2} is also adjacent to a leaf, a perfect matching must match v_{0} and v_{2} to their adjacent leaves. But then vertex v_{1} with neighbors v_{0} and v_{2} cannot be matched. Hence v_{2} is not adjacent to a leaf.
- i.
If v_{2} has degree at most two in T, then Operation 4 is applicable to the tree extending path (v_{0},v_{1},v_{2}). The edge {v_{1},v_{2}} is contained in the perfect matching of T: vertex v_{0} can only be matched to its adjacent leaf, and since v_{1} has degree two the only remaining edge incident on it which can be in a matching is indeed {v_{1},v_{2}}.
- ii.
If v_{2} has degree at least three in T, then since we derived earlier that v_{2} is not adjacent to a leaf we find that Operation 5 is applicable to the tree extending path (v_{0},v_{1},v_{2}).
- i.
- (a)
- 3.If deg_{T}(v_{0})=2, let v_{1} be the unique neighbor of v_{0} not contained in T′ which exists by definition of an open branch vertex. Consider the maximal path P=(v_{0},v_{1},…,v_{q}) obtained by starting with the edge {v_{0},v_{1}} and following degree-2 vertices until arriving at the first vertex v_{q} which has degree unequal to two in T.
- (a)
If (a) deg_{T}(v_{1})≥4 or (b) deg_{T}(v_{1})=3 and v_{1} is not adjacent to a leaf, then Operation 5 is applicable to the tree extending path (v_{0},v_{1}).
- (b)
If deg_{T}(v_{1})=3 and v_{1} is adjacent to a leaf, then v_{1} is a spike vertex which shows that Operation 2 is applicable.
- (c)
If deg_{T}(v_{1})=1 then Operation 4 is applicable to the extending path (v_{0},v_{1}).
- (d)In the remainder we therefore have deg_{T}(v_{1})=2, which implies by the definition of the path P we are considering that there is a vertex v_{2}.
- i.
If deg_{T}(v_{2})≤2 then we claim Operation 4 is applicable. Since the degree of v_{1} in T is two, either the edge {v_{0},v_{1}} or {v_{1},v_{2}} is contained in the perfect matching, which shows that the mentioned operation can be applied to the path (v_{0},v_{1}) or (v_{0},v_{1},v_{2}) depending on which case holds.
- ii.
In the remainder we therefore have deg_{T}(v_{2})≥3. If v_{2} is adjacent to a leaf, then v_{2} must be matched to this leaf in the perfect matching which shows that v_{1} is matched to v_{0}: hence Operation 4 is applicable to (v_{0},v_{1}).
- iii.
If v_{2} is not adjacent to a leaf, then since its degree is at least three we find that Operation 5 is applicable to the tree extending path (v_{0},v_{1},v_{2}).
- i.
- (a)
We remark that by using a more detailed case analysis, one could prove better bounds for the number of conflict structures that can be found in a tree with a perfect matching. An improvement in the bound immediately leads to a better provable upper bound on the kernel size. But since such an improvement does not actually decrease the size of reduced instances (it only affects what we can prove about the sizes of such instances — it does not affect what any of the reduction rules do), and would not change the cubic dependency of the kernel size on the parameter, we have chosen not to pursue this bound further in the interest of space and readability.
3.4 The Kernelization Algorithm
Using the packing argument from the previous section, we can finally prove an upper bound on the size of reduced instances.
Lemma 8
Let (G,X,k) be a reduced clean instance offvs-Independent Setwith forest F:=G−X. Then \(|V(G)| \leq|X| + 14|X|(|X| + \binom{|X|}{2})\).
Proof
Consider such a reduced instance. By definition of the instance being clean, the forest F has a perfect matching. By applying Theorem 1 to each tree in the forest F, we find obtain a set \(\mathcal{S}\) of vertex-disjoint conflict structures in F with \(|\mathcal{S}| \geq|V(F)| / 14\). By Lemma 7 this shows that \(\mathop{\mathrm {\mbox{\textsc{Active}}}}_{F}(\mathcal{X}) \geq |V(F)| / 14\). On the other hand, Observation 4 gives the bound \(\mathop{\mathrm {\mbox{\textsc{Active}}}}_{F}(\mathcal{X}) \leq |X|^{2} + \binom{|X|}{2}|X|\). We therefore find that \(|V(F)| \leq 14(|X|^{2} + \binom{|X|}{2}|X|)\). Since |V(G)|=|X|+|V(F)| we conclude that \(|V(G)| \leq|X| + 14|X|(|X| + \binom{|X|}{2})\). □
The previous lemma gives a size bound for reduced instances. Before proving the existence of a kernel using this bound, let us consider how much time is needed to compute a reduced instance.
Lemma 9
Given a clean instance (G,X,k) offvs-Independent Seton nvertices we can exhaustively apply Reduction Rules 1–5 in \(\mathcal{O}(|X|^{2} \cdot n)\)time to output an equivalent reduced instance (G′,X′,k′).
Proof
The crucial idea is to apply the reduction rules in a suitable order, to prevent re-triggering reduction rules which were already applied before. This will ensure that we need only a single pass over the instance to exhaustively reduce it, which improves the running time.
Start by computing for each chunk \(Y \in\mathcal {X}\) the value \(\mathop{\mathrm{\mbox{\textsc {Conf}}}}_{F}(Y)\). Since we can precompute the value α(F) once in linear-time, for each choice of Y we can compute \(\mathop{\mathrm{\mbox{\textsc {Conf}}}}_{F}(Y)\) in \(\mathcal{O}(|V(F)|)\) time by marking which vertices of F are adjacent to Y, finding a MIS among the vertices of F which are not marked, and comparing its size to the precomputed value. Now bucket-sort the chunks based on the number of conflicts they induce; since the number of conflicts is at most |V(F)| we can bucket-sort in \(|V(F)| + |\mathcal {X}|\) time. Then consider the chunks in decreasing value of the number of conflicts they induce and apply Rules 1 and twoVertexDeletion where possible, using the current size of the feedback vertex set when testing for applicability. Observe that an application of Rule 1 might decrease the size of the feedback vertex set X, which could cause a rule to become applicable for other chunks where it was not applicable before. By treating chunks in order of decreasing conflict value and testing for applicability of a rule when handling a chunk, we ensure that reduction rules do not become applicable to chunks we have already considered — observe that the number of conflicts induced by a chunk does not change when applying Rules 1 or 2 elsewhere, except when deleting a vertex involved in some chunk (which can be handled easily). Hence after doing one such pass over the instance in \(\mathcal{O}(|\mathcal {X}| \cdot|V(F)|) \subseteq\mathcal{O}(|X|^{2} \cdot n)\) time, we end up with an equivalent instance (G_{1},X_{1},k_{1}) to which Rule 1 and Rule 2 do not apply.
As the next phase we will exhaustively apply Rule 4 and Rule 5. The crucial fact we use here is that an application of one of these two rules does not change the number of conflicts that is induced by any chunk \(Y \in\mathcal{X}\), which can be proven by arguments similar to those used to argue the correctness of the two reduction rules. Hence by applying Rules 4 and 5 we do not change the fact that the instance is reduced with respect to Rules 1 and 2. It is not hard to see that a forest F can contain at most \(\mathcal{O}(|V(F)|)\) structures which satisfy the degree constraints of Rules 4 and 5, which follows from the fact that F is acyclic and the relevant substructures are subgraphs of constant degree. We may identify all these structures in \(\mathcal{O}(|V(F)|)\) time by using a suitable depth-first search; we omit the straight-forward details of such a procedure. For each substructure to which Rule 4 might be applied (an edge whose endpoints have degree at most two), or to which Rule 5 might be applied (four vertices on a path with degrees one, three, three, and one), we can test whether a rule is applicable in \(\mathcal{O}(|X|^{2})\) time: the effort here lies in testing whether a pair of vertices u,v from F is X-blockable. Using an adjacency-matrix for X we can test for each vertex in N_{G}(u)∩X whether it is adjacent to all vertices in N_{G}(v)∩X; the pair is X-blockable if and only if this is false. Once we determine that a rule is applicable, we modify the graph as needed. This involves modifying constant-degree constant-size substructures in F, which have arbitrary adjacencies to X. Using an appropriate data-structure such as an adjacency-list, we may perform these local modifications in \(\mathcal{O}(|X|)\) time. By applying Rule 4 we might trigger Rule 5, or vice versa. Luckily, we can only trigger a rule which was not applicable before in the immediate neighborhood of the previous structure which was reduced, and we can test whether this happens in constant time. Since each reduction rule decreases the number of vertices in F, we apply the rules at most |V(F)| times. Each application can be performed in \(\mathcal{O}(|X|^{2})\) time. By using a suitable depth-first search we can identify all structures which satisfy the degree constraints of the two rules in \(\mathcal{O}(|V(F)|)\) time. In total we therefore find that from (G_{1},X_{1},k_{1}) we may compute an equivalent instance (G_{2},X_{2},k_{2}) which is reduced with respect to Rules 1, 2, 4 and 5 in \(\mathcal{O}(|X|^{2} \cdot|V(F)|)\) time.
As the final step, the algorithm needs to apply Rule 3. It is trivial to verify that this rule does not trigger any other reduction rules. We may apply this rule by computing for each chunk \(Y \in\mathcal{X}\), for each remaining tree T in the forest, the number of conflicts induced on T by Y in a manner similar as described before. Afterwards we delete trees for which no chunks induce a conflict. This phase is easily implemented to run in \(\mathcal{O}(|\mathcal{X}| \cdot|V(F)|)\) time. We output the resulting instance (G_{3},X_{3},k_{3}) of the problem, which was found in \(\mathcal{O}(|X|^{2} \cdot n)\) time overall. □
Theorem 2
fvs-Independent Sethas a kernel with a cubic number of vertices: there is an algorithm that transforms an instance (G,X,k) on nvertices and medges into an equivalent instance (G′,X′,k′) in\(\mathcal {O}(\sqrt {n} m + n^{5/3})\)time such that |X′|≤2|X| and |V(G′)|≤2|X|+28|X|^{2}+56|X|^{3}.
Proof
Given an input instance (G,X,k) of fvs-Independent Set, we first apply Lemma 1 to obtain an equivalent clean instance (G_{1},X_{1},k_{1}) with |X_{1}|≤2|X| in \(\mathcal{O}(\sqrt{n}m)\) time. To optimize the running time of the kernelization algorithm, we do not further process the instance if n≤|X|^{3}, but simply output (G_{1},X_{1},k_{1}) as the result of the procedure; this is suitably small since |V(G_{1})|≤|V(G)|≤|X|^{3} in this case.
In the remainder we may therefore assume that n>|X|^{3}, which implies |X_{1}|≤2⋅n^{1/3} as |X_{1}|≤2|X|. We invoke Lemma 9 to obtain an equivalent reduced instance (G_{2},X_{2},k_{2}) in \(\mathcal{O}(|X_{1}|^{2} \cdot|V(G_{1})|)\subseteq \mathcal{O}(n^{2/3} \cdot n)\) time. Since the reduction rules do not change the fact that the instance is clean, the reduced instance is also clean. By Lemma 8 the size of the resulting graph G_{2} is bounded by \(|V(G_{2})| \leq|X_{2}| +14|X_{2}|(|X_{2}| + \binom{|X_{2}|}{2})\). As the reduction rules do not increase the size of the feedback vertex set, we have |X_{2}|≤|X_{1}|≤2|X|. We therefore obtain |V(G_{2})|≤2|X|+28|X|^{2}+56|X|^{3} by plugging in the bound on X_{2} and evaluating the binomial expression. We output the instance (G_{2},X_{2},k_{2}) as the result of the kernelization, or a trivial yes-instance if k_{2}≤0. By the correctness of the reduction rules, this instance is equivalent to the input instance. The set X_{2} is a feedback vertex set for G_{2}, since X_{1} is a FVS for G_{1} and the reduction rules preserve this. Observe that the original set X (or what is left of it in the final graph G_{2}) might not constitute a FVS for G_{2}, as edges may have been added between vertices which were added to the feedback vertex set in order to clean the instance. The running time of the procedure is \(\mathcal{O}(\sqrt{n} m + n^{5/3})\). □
Using the previous theorem we easily obtain a corollary about kernelization for Vertex Cover from its relationship to Independent Set.
Corollary 1
fvs-Vertex Coverhas a kernel with min(2k,2|X|+28|X|^{2}+56|X|^{3}) vertices which can be computed in \(\mathcal{O}(\sqrt{n} m + n^{5/3})\)time.
Proof
Given an instance (G,X,k) of fvs-Vertex Cover we transform it into an instance (G,X,|V(G)|−k) of fvs-Independent Set, which is an equivalent instance because the complement of an independent set is a vertex cover. We apply the kernelization algorithm from Theorem 2 to (G,X,|V(G)|−k) to compute in \(\mathcal{O}(\sqrt {n} m + n^{5/3})\) time an equivalent instance (G′,X′,|V(G′)|−k′). By adjusting the target value we transform this back to an instance (G′,X′,k′) of fvs-Vertex Cover and use it as the output, which shows that |V(G′)|≤2|X|+28|X|^{2}+56|X|^{3}. Since the kernelization for fvs-Independent Set starts by applying the Nemhauser-Trotter decomposition which is known to yield a 2k-vertex kernel [9], the number of vertices in the resulting graph G′ is also bounded by 2k, where k is the size of the vertex cover that is asked for by the original input instance. □
We remark that the Vertex Cover kernelization with respect to the parameter \(\mathop{\mathrm{\mbox{\textsc{fvs}}}}(G)\) can be combined with any existing Vertex Cover kernel which reduces the graph by only deleting vertices. Since all existing Vertex Cover kernels (the Buss rule [7], crown reductions [2, 11, 12] and the Nemhauser-Trotter reduction [9, 35]) are of this type, our reduction rules can be combined with all of these.
4 No Polynomial Kernel for VC-Weighted Vertex Cover
The goal of this section is to prove that vertex weights make it much harder to kernelize an instance of the vertex cover problem. To prove a kernelization lower bound for vc-Weighted Vertex Cover we use the recently introduced notion of cross-composition [5] which builds on earlier work by Bodlaender et al. [4], and Fortnow and Santhanam [25].
Definition 10
(Polynomial equivalence relation [5])
- 1.
There is an algorithm that given two strings x,y∈Σ^{∗} decides whether x and y belong to the same equivalence class in \((|x| + |y|)^{\mathcal{O}(1)}\) time.
- 2.
For any finite set S⊆Σ^{∗} the equivalence relation \(\mathcal{R}\) partitions the elements of S into at most \((\max _{x \in S} |x|)^{\mathcal{O}(1)}\) classes.
Definition 11
(Cross-composition [5])
- 1.
(x^{∗},k^{∗})∈Q⇔x_{i}∈L for some 1≤i≤t,
- 2.
k^{∗} is bounded by a polynomial in \(\max_{i=1}^{t} |x_{i}|+\log t\).
Theorem 3
[5]
If some set L⊆Σ^{∗}is NP-hard under Karp reductions and Lcross-composes into the parameterized problem Qthen there is no polynomial kernel for Qunless NP ⊆ coNP/poly.
The following proposition will enable us to establish the NP-completeness of Independent Set onP_{2}-Split Graphs. It is the reverse of the “folding rule” which was used for vertex cover kernelization by Chen et al. [9, Lemma 2.3].Independent Set onP_{2}-Split Graphs
Instance: A graph G, an independent set Y in G such that each component of G−Y is isomorphic to P_{2}, and an integer k.
Question: Does G have an independent set of size at least k?
Proposition 2
Let Gbe a graph and let {u,v}∈E(G). Let G′ be the graph obtained from Gby removing the edge {u,v}, adding two new vertices p_{1},p_{2}and the edges {u,p_{1}},{p_{1},p_{2}},{p_{2},v}. Then α(G′)=α(G)+1.
Lemma 10
Independent Set onP_{2}-Split Graphsis NP-complete.
Proof
Membership in NP is trivial; we prove hardness by a reduction from the unrestricted Independent Set problem [26, GT20]. Consider an instance (G,k) of Independent Set. Now obtain a graph G′ by replacing each edge {u,v}∈E(G) by a path on two new vertices whose endpoints are adjacent to u and v, respectively. If we let Y:=V(G) be the set of original vertices in the graph G′ then using Proposition 2 it is not hard to see that instance (G′,Y,k+|E(G)|) is equivalent to (G,k), which concludes the proof. □
We can prove a kernelization lower bound for this problem using cross-composition.vc-Weighted Independent Set
Instance: A simple undirected graph G, a weight function \(w: V(G) \to\mathbb{N^{+}}\), a vertex cover X⊆V(G), an integer k≥0.
Parameter: The cardinality |X| of the vertex cover.
Question: Is there an independent set C of G such that ∑_{v∈C}w(v)≥k?
Theorem 4
vc-Weighted Independent Setdoes not admit a polynomial kernel unless NP ⊆ coNP/poly.
Proof
By Theorem 3 and Lemma 10 it is sufficient to prove that Independent Set onP_{2}-Split Graphs cross-composes into vc-Weighted Independent Set. We start by defining a suitable polynomial equivalence relationship \(\mathcal {R}\). Fix some reasonable encoding of instances of Independent Set onP_{2}-Split Graphs into strings on an alphabet Σ. Now let two strings x,y∈Σ^{∗} be equivalent under \(\mathcal{R}\) if (a) both strings do not encode a well-formed instance of Independent Set onP_{2}-Split Graphs, or (b) the strings encode instances (G_{1},Y_{1},k_{1}) and (G_{2},Y_{2},k_{2}) such that |V(G_{1})|=|V(G_{2})|,|Y_{1}|=|Y_{2}| and k_{1}=k_{2}. It is not difficult to see that a set of strings which encodes instances on at most n vertices each, is partitioned into \(\mathcal{O}(n^{3})\) equivalence classes. A reasonable encoding of input instances allows equivalence to be tested in polynomial time, and hence \(\mathcal{R}\) is a polynomial equivalence relationship according to Definition 10.
We now give an algorithm that receives t instances of Independent Set onP_{2}-Split Graphs which are equivalent under \(\mathcal{R}\), and constructs an instance of vc-Weighted Independent Set with small parameter value that acts as the OR of the inputs. If the input instances are not well-formed, then we simply output a constant-sized no-instance. Using the properties of \(\mathcal{R}\) we may therefore assume in the remainder that the input instances are (G_{1},Y_{1},k_{1}),…,(G_{t},Y_{t},k_{t}) such that |V(G_{1})|=⋯=|V(G_{t})|=n, |Y_{1}|=⋯=|Y_{t}|=r and k_{1}=⋯=k_{t}=k. We may assume without loss of generality (by duplicating some instances if needed) that t is a power of two. We construct an instance (G′,w′,X′,k′) of vc-Weighted Independent Set as follows.
- 1.
Initialize G′ as the disjoint union \(\dot{\bigcup}_{i=1}^{t} Y_{i}\) of the independent sets of the input instances. Set the weight of all these vertices to one.
- 2.For j∈[q] add vertices \(a'_{j}, b'_{j}\) of weight one and the edge \(\{a'_{j}, b'_{j}\}\) to G′. Connect these vertices to the other vertices as follows.
For i∈[t], for each vertex v∈Y_{i} and for each j∈[q] make v adjacent to \(a'_{j}\) (resp. \(b'_{j}\)) if and only if v is adjacent in G_{i} to a_{j} (resp. b_{j}).
- 3.For j∈[logt] add vertices \(s^{0}_{j}, s^{1}_{j}\) to G′ of weight t(n+1) each, and add the edge \(\{s^{0}_{j}, s^{1}_{j}\}\). Connect these to the remainder of the graph as follows.
For i∈[t] and j∈[logt], let b(i,j)∈{0,1} be the value of the j-bit in the binary expansion of number i. Make vertex \(s^{b(i,j)}_{j}\) adjacent to all vertices of Y_{i} that were added to G′ in the first step.
This concludes the construction of the weighted graph (G′,w′). Observe the important fact that for each i∈[t] the graph \(G'[ Y_{i}\cup\{ a'_{j}, b'_{j} \mid j \in[q] \} ]\) is isomorphic to G_{i} since the vertices of Y_{i} form an independent set in both graphs, the remaining vertices form a disjoint union of P_{2}’s in both graphs, and the connections between the vertices of Y_{i} and the P_{2}’s are identical.
We set k′:=k+t(n+1)logt. Define \(X' := \{ a'_{j}, b'_{j} \mid j\in[q]\} \cup\{ s^{0}_{j}, s^{1}_{j} \mid j \in[\log t] \}\). Since the only vertices of G′−X′ are the vertices corresponding to the sets Y_{i} of the input instances, and since we have not added any edges between these vertices, it follows that G′−X′ is an independent set and therefore X′ is a vertex cover of size |X′|=2q+2logt≤2n+2logt. Since the size of an input instance on n vertices is at least n bits in a reasonable encoding (under which Independent Set onP_{2}-Split Graphs is NP-complete), it follows that |X′| is bounded by a polynomial in the size of the largest input instance plus logt. We can therefore output the instance (G′,w′,X′,k′) of vc-Weighted Independent Set, knowing that the parameter value |X′| is suitably bounded. It remains to prove that this output instance is yes if and only if one of the input instances is yes.
For the first direction, assume that (G′,w′,X′,k′) is yes and let C⊆V(G′) be an independent set of total weight at least k′. Since \(\{s^{0}_{j}, s^{1}_{j}\} \in E(G')\) for j∈[logt], the independent set C contains at most one vertex of each pair \(s^{0}_{j}, s^{1}_{j}\). Since the only vertices which have weight unequal to one are the s-vertices of weight t(n+1), and since the number of vertices in G′ which are not s-vertices is at most t⋅n, it follows that if C does not contain one vertex of each pair \(s^{0}_{j},s^{1}_{j}\) (j∈[logt]) then the weight of C is at most t(n+1)(logt−1)+t⋅n≤t(n+1)logt−t<k′. Hence by the assumption that C has weight at least k′ we know that C contains exactly one vertex of each pair \(s^{0}_{j}, s^{1}_{j}\). Consider the number i^{∗} whose j-th bit is 1 if \(s^{0}_{j}\) is in C, and whose j-th bit is 0 otherwise. For all instance numbers i≠i^{∗} the binary representation of i disagrees with the binary representation of i^{∗} on at least one position j∈[logt], which implies by the construction of G′ that all vertices of Y_{i} in G′ are adjacent to the vertex of \(s^{0}_{j}, s^{1}_{j}\) which is contained in C. Since C is an independent set, this shows that C does not contain any vertices of Y_{i} for all i≠i^{∗}. Since the weight of C is at least k+t(n+1)logt and C contains exactly one vertex of each s-pair, this shows that the vertices of \(Y_{i^{*}} \cup\{ a'_{j},b'_{j} \mid j \in[q] \}\) must contribute at least k to the weight of C. Since each vertex in this latter set has weight one, this shows that \(|C \cap( Y_{i^{*}} \cup\{ a'_{j}, b'_{j} \mid j \in[q] \} )| \geq k\). But by construction of G′ we know that \(G'[ Y_{i^{*}} \cup\{ a'_{j},b'_{j} \mid j \in[q] \} ]\) is isomorphic to \(G_{i^{*}}\) and therefore \(G_{i^{*}}\) contains an independent set of size at least k, proving that input instance \((G_{i^{*}}, Y_{i^{*}}, k_{i^{*}})\) is yes.
For the reverse direction, assume that \((G_{i^{*}}, Y_{i^{*}}, k_{i^{*}})\) is a yes-instance; we prove that the constructed instance contains an independent set of weight k′. Since \(G'[ Y_{i^{*}} \cup\{ a'_{j}, b'_{j}\mid j \in[q] \}]\) is isomorphic to \(G_{i^{*}}\), this induced subgraph contains an independent set C of cardinality k and hence of weight k. Now consider the binary expansion of the number i^{∗}. For j∈[logt] if the j-th bit of i^{∗} is 1, then vertex \(s^{0}_{j}\) is not adjacent to any of the vertices in C and hence can be added to C without violating independence; if the bit is 0 then \(s^{1}_{j}\) can be added. Since vertices s_{j} for different values of j are not adjacent to each other, we can add one vertex of each pair \(s^{0}_{j}, s^{1}_{j}\) to C in this fashion for j∈[logt] to obtain an independent set of weight k+t(n+1)logt which proves that the output instance is yes.
This concludes the proof that the constructed instance is equivalent to the OR of the input instances. Since the construction can be carried out in polynomial time this is a valid cross-composition, and by Theorem 3 this concludes the proof. □
Corollary 2
vc-Weighted Vertex Coverdoes not admit a polynomial kernel unless NP ⊆ coNP/poly.
Proof
Since an instance (G,w,X,k) of vc-Weighted Independent Set is equivalent to an instance (G,w,X,(∑_{v∈V(G)}w(v))−k) of vc-Weighted Vertex Cover with the same parameter, the construction of Theorem 4 also shows that Independent Set onP_{2}-Split Graphs cross-composes into vc-Weighted Vertex Cover which proves the claim. □
5 Conclusion
We have given a cubic kernel for the Vertex Cover and Independent Set problems using the parameter \(\mathop{\mathrm{\mbox{\textsc {fvs}}}}(G)\). It would be very interesting to perform experiments with our new reduction rules to see whether they offer significant benefits over the existing Vertex Cover kernel on real-world instances. This result is one of the first examples of a polynomial kernel using a “refined” parameter which is structurally smaller than the standard parameterization. The kernel we have presented for fvs-Vertex Cover contains \(\mathcal{O}(|X|^{3})\) vertices. Since a graph G with feedback vertex set X has at most \(\binom{|X|}{2} +|V(G) \setminus X| \cdot|X| + |V(G) \setminus X| - 1\) edges, a reduced instance can be encoded in \(\mathcal{O}(|X|^{4} \log|X|)\) bits using an adjacency-list since an adjacency-list encoding of a graph takes \(\mathcal{O}(\log|V(G)| + |E(G)| \log|V(G)|)\) bits. The results of Dell and Van Melkebeek [15] imply that it is unlikely that there exists a kernel which can be encoded in \(\mathcal{O}(|X|^{2 - \epsilon})\) bits for any ϵ>0. It might be possible to improve the size of the kernel to a quadratic or even a linear number of vertices, by employing new reduction rules. The current reduction rules can be seen as analogs of the traditional “high degree” rule for the Vertex Cover problem, and it would be interesting to see whether it is possible to find analogs of crown reduction rules when using \(\mathop{\mathrm{\mbox {\textsc{fvs}}}}(G)\) as the parameter.
Although we have assumed throughout the paper that a feedback vertex set is supplied with the input, we can drop this restriction by applying the known polynomial-time 2-approximation algorithm for FVS [3]. Observe that the reduction algorithm does not require that the supplied set X is a minimum feedback vertex set; the kernelization algorithm works if X is any feedback vertex set, and the size of the output instance depends on the size of the FVS that is supplied. Hence if we compute a 2-approximate FVS and use it in the kernelization algorithm, the bound on the number of vertices in the output instance is only a factor 8 worse than when running the kernelization using a minimum FVS.
This paper has focused on the decision version of the Vertex Cover problem, but the data reduction rules given here can also be translated to the optimization version to obtain the following result: given a graph G there is a polynomial-time algorithm that computes a graph G′ and a non-negative integer c such that \(\mathop{\mathrm {\mbox{\textsc{vc}}}}(G) = \mathop{\mathrm{\mbox{\textsc{vc}}}}(G')+ c\) with \(|V(G')| \leq2\mathop{\mathrm{\mbox{\textsc{vc}}}}(G)\) and \(|V(G')| \in\mathcal{O}(\mathop{\mathrm{\mbox{\textsc{fvs}}}}(G)^{3})\); and a vertex cover S′ for G′ can be transformed back into a vertex cover of G of size |S′|+c in polynomial time.
Weighted Problems
In Sect. 4 we proved that the vc-Weighted Vertex Cover problem does not admit a polynomial kernel unless NP ⊆ coNP/poly. Of course this immediately implies a kernel lower bound for the weighted problem parameterized by the size of a feedback vertex set. After the preliminary version of this paper appeared (where we proved the lower bound for fvs-Weighted Vertex Cover), we have found several other weighted problems parameterized by the cardinality of a given vertex cover which are FPT but do not admit polynomial kernels unless NP ⊆ coNP/poly, including Weighted Feedback Vertex Set [5], Weighted Treewidth [6] and Weighted Odd Cycle Transversal [31]. It seems that for problems parameterized by the size of a given vertex cover, the presence of vertex weights forms an obstruction to the existence of polynomial kernels. This trend can be compared to the observation that for vertex- or edge subset problems under the natural parameterization, the presence of connectivity requirements often excludes a polynomial kernelization. For example, well-known connectivity problems without polynomial kernels include k-Path [4], k-Connected Vertex Cover [17] and k-Connected Feedback Vertex Set [14] (assuming NP \(\not\subseteq\) coNP/poly). Uncovering further properties of problems which are strongly correlated to the existence of polynomial kernels seems like an interesting area of further research.
Other Parameterizations
The approach of studying Vertex Cover parameterized by \(\mathop{\mathrm{\mbox {\textsc{fvs}}}}(G)\) fits into the broad context of “parameterizing away from triviality” [8, 36], since the parameter \(\mathop{\mathrm{\mbox{\textsc{fvs}}}}(G)\) measures how many vertex-deletions are needed to reduce G to a forest in which Vertex Cover can be solved in polynomial time. As there is a wide variety of restricted graph classes for which Vertex Cover is in P, this opens up a multitude of possibilities for non-standard parameterizations. As observed by Cai [8], for every graph class \(\mathcal{G}\) which is closed under vertex deletion and for which the Vertex Cover problem is in P, the Vertex Cover problem is in FPT when parameterized by the size of a set X such that \(G - X \in\mathcal{G}\), assuming that X is given as part of the input. Such problems can be solved in \(\mathcal{O}^{*}(2^{|X|})\) time by enumerating all independent subsets X′⊆X and computing α(G−X−N_{G}(X′)), which can be done in polynomial time since \(G - X - N_{G}(X') \in\mathcal{G}\). The independence number of G is the maximum of |X′|+α(G−X−N_{G}(X′)) over all independent subsets X′. In the recent paper on cross-composition [5], a superset of the authors showed that whenever \(\mathcal{G}\) contains all cliques the resulting parameterized problem does not have a polynomial kernel unless NP ⊆ coNP/poly. This implies that for classes such as claw-free graphs, interval graphs and various other types of perfect graphs, Vertex Cover parameterized by the size of a given deletion set to the class is in FPT, but does not admit a polynomial kernel unless NP ⊆ coNP/poly. Further research may try to find more general graph classes \(\mathcal {G}\) such that Vertex Cover admits a polynomial kernel parameterized by deletion distance to \(\mathcal{G}\). Since relevant candidate classes cannot contain arbitrarily large cliques and must admit polynomial-time algorithms for solving Vertex Cover, bipartite graphs might be an interesting subject for further study.
One might also consider the Vertex Cover problem parameterized by the size of a given set X such that \(\mathop{\mathrm{\mbox{\textsc {Treewidth}}}}(G - X) \leq i\). The classic Vertex Cover kernelizations can be interpreted as the case i=0, whereas this paper supplies the result for i=1. It was recently proven that the positive results cannot extend further in this direction: Cygan et al. [13] showed that the case i=2 does not admit a polynomial kernel unless NP ⊆ coNP/poly.
Footnotes
- 1.
We use k-Vertex Cover to denote the parameterization by the target size k.
Notes
Acknowledgements
We are grateful to the anonymous referees, whose suggestions significantly improved the exposition of our results and decreased the running time of the kernelization procedure.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
References
- 1.Abu-Khzam, F.N., Collins, R.L., Fellows, M.R., Langston, M.A., Suters, W.H., Symons, C.T.: Kernelization algorithms for the vertex cover problem: theory and experiments. In: Proc. 6th ALENEX/ANALC, pp. 62–69 (2004) Google Scholar
- 2.Abu-Khzam, F.N., Fellows, M.R., Langston, M.A., Suters, W.H.: Crown structures for vertex cover kernelization. Theory Comput. Syst. 41(3), 411–430 (2007). doi:10.1007/s00224-007-1328-0 MathSciNetMATHCrossRefGoogle Scholar
- 3.Bafna, V., Berman, P., Fujito, T.: A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM J. Discrete Math. 12(3), 289–297 (1999). doi:10.1137/S0895480196305124 MathSciNetMATHCrossRefGoogle Scholar
- 4.Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009). doi:10.1016/j.jcss.2009.04.001 MathSciNetMATHCrossRefGoogle Scholar
- 5.Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Cross-composition: a new technique for kernelization lower bounds. In: Proc. 28th STACS, pp. 165–176 (2011). doi:10.4230/LIPIcs.STACS.2011.165 Google Scholar
- 6.Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Preprocessing for treewidth: a combinatorial analysis through kernelization. In: Proc. 38th ICALP, pp. 437–448 (2011). doi:10.1007/978-3-642-22006-7_37 Google Scholar
- 7.Buss, J.F., Goldsmith, J.: Nondeterminism within P. SIAM J. Comput. 22(3), 560–572 (1993). doi:10.1137/0222038 MathSciNetMATHCrossRefGoogle Scholar
- 8.Cai, L.: Parameterized complexity of vertex colouring. Discrete Appl. Math. 127(3), 415–429 (2003). doi:10.1016/S0166-218X(02)00242-1 MathSciNetMATHCrossRefGoogle Scholar
- 9.Chen, J., Kanj, I.A., Jia, W.: Vertex cover: further observations and further improvements. J. Algorithms 41(2), 280–301 (2001). doi:10.1006/jagm.2001.1186 MathSciNetMATHCrossRefGoogle Scholar
- 10.Chen, J., Kanj, I.A., Xia, G.: Improved upper bounds for vertex cover. Theor. Comput. Sci. 411(40–42), 3736–3756 (2010). doi:10.1016/j.tcs.2010.06.026 MathSciNetMATHCrossRefGoogle Scholar
- 11.Chlebík, M., Chlebíková, J.: Crown reductions for the minimum weighted vertex cover problem. Discrete Appl. Math. 156(3), 292–312 (2008). doi:10.1016/j.dam.2007.03.026 MathSciNetMATHCrossRefGoogle Scholar
- 12.Chor, B., Fellows, M., Juedes, D.W.: Linear kernels in linear time, or how to save k colors in O(n ^{2}) steps. In: Proc. 30th WG, pp. 257–269 (2004). doi:10.1007/978-3-540-30559-0_22 Google Scholar
- 13.Cygan, M., Lokshtanov, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: On the hardness of losing width. In: Proc. 6th IPEC (2012) (To appear) Google Scholar
- 14.Cygan, M., Pilipczuk, M., Pilipczuk, M., Wojtaszczyk, J.O.: Kernelization hardness of connectivity problems in 2-degenerate graphs. In: Proc. 36th WG, pp. 147–158 (2010). doi:10.1007/978-3-642-16926-7_15 Google Scholar
- 15.Dell, H., van Melkebeek, D.: Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. In: Proc. 42nd STOC, pp. 251–260 (2010). doi:10.1145/1806689.1806725 Google Scholar
- 16.Díaz, J., Petit, J., Thilikos, D.M.: Kernels for the vertex cover problem on the preferred attachment model. In: Proc. 5th WEA, pp. 231–240 (2006). doi:10.1007/11764298_21 Google Scholar
- 17.Dom, M., Lokshtanov, D., Saurabh, S.: Incompressibility through colors and IDs. In: Proc. 36th ICALP, pp. 378–389 (2009). doi:10.1007/978-3-642-02927-1_32 Google Scholar
- 18.Downey, R., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, New York (1999) CrossRefGoogle Scholar
- 19.Downey, R.G., Fellows, M.R., Langston, M.A. (eds.): Comput. J.: Special Issue on Parameterized Complexity, 51 (2008) Google Scholar
- 20.Downey, R.G., Fellows, M.R., Stege, U.: Parameterized complexity: a framework for systematically confronting computational intractability. In: DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pp. 49–99 (1997) Google Scholar
- 21.Estivill-Castro, V., Fellows, M., Langston, M., Rosamond, F.: FPT is P-time extremal structure I. In: Proc. 1st ACiD, pp. 1–41 (2005) Google Scholar
- 22.Fellows, M.R.: Towards fully multivariate algorithmics: some new results and directions in parameter ecology. In: Proc. 20th IWOCA, pp. 2–10 (2009). doi:10.1007/978-3-642-10217-2_2 Google Scholar
- 23.Fellows, M.R., Lokshtanov, D., Misra, N., Mnich, M., Rosamond, F.A., Saurabh, S.: The complexity ecology of parameters: an illustration using bounded max leaf number. Theory Comput. Syst. 45(4), 822–848 (2009). doi:10.1007/s00224-009-9167-9 MathSciNetMATHCrossRefGoogle Scholar
- 24.Fellows, M.R., Lokshtanov, D., Misra, N., Rosamond, F.A., Saurabh, S.: Graph layout problems parameterized by vertex cover. In: Proc. 19th ISAAC, pp. 294–305 (2008). doi:10.1007/978-3-540-92182-0_28 Google Scholar
- 25.Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci. 77(1), 91–106 (2011). doi:10.1016/j.jcss.2010.06.007 MathSciNetMATHCrossRefGoogle Scholar
- 26.Garey, M.R., Johnson, D.S.: Computers and Intractability, a Guide to the Theory of NP-Completeness. Freeman, New York (1979) MATHGoogle Scholar
- 27.Griggs, J.R., Kleitman, D., Shastri, A.: Spanning trees with many leaves in cubic graphs. J. Graph Theory 13, 669–695 (1989). doi:10.1002/jgt.3190130604 MathSciNetMATHCrossRefGoogle Scholar
- 28.Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007). doi:10.1145/1233481.1233493 CrossRefGoogle Scholar
- 29.Gutin, G., Kim, E.J., Lampis, M., Mitsou, V.: Vertex cover problem parameterized above and below tight bounds. Theory Comput. Syst. 48(2), 402–410 (2011). doi:10.1007/s00224-010-9262-y MathSciNetMATHCrossRefGoogle Scholar
- 30.Jansen, B.M.P., Bodlaender, H.L.: Vertex cover kernelization revisited: upper and lower bounds for a refined parameter. In: Proc. 28th STACS, pp. 177–188 (2011). doi:10.4230/LIPIcs.STACS.2011.177 Google Scholar
- 31.Jansen, B.M.P., Kratsch, S.: On polynomial kernels for structural parameterizations of odd cycle transversal. In: Proc. 6th IPEC (2012) (To appear) Google Scholar
- 32.Khot, S., Regev, O.: Vertex cover might be hard to approximate to within 2−ϵ. J. Comput. Syst. Sci. 74(3), 335–349 (2008). doi:10.1016/j.jcss.2007.06.019 MathSciNetMATHCrossRefGoogle Scholar
- 33.Kratsch, S., Schweitzer, P.: Isomorphism for graphs of bounded feedback vertex set number. In: Proc. 12th SWAT, pp. 81–92 (2010). doi:10.1007/978-3-642-13731-0_9 Google Scholar
- 34.Mishra, S., Raman, V., Saurabh, S., Sikdar, S., Subramanian, C.: The complexity of König subgraph problems and above-guarantee vertex cover. Algorithmica 61(4), 857–881 (2011). doi:10.1007/s00453-010-9412-2 MathSciNetMATHCrossRefGoogle Scholar
- 35.Nemhauser, G., Trotter, L.: Vertex packings: structural properties and algorithms. Math. Program. 8, 232–248 (1975). doi:10.1007/BF01580444 MathSciNetMATHCrossRefGoogle Scholar
- 36.Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006) MATHCrossRefGoogle Scholar
- 37.Niedermeier, R.: Reflections on multivariate algorithmics and problem parameterization. In: Proc. 27th STACS, pp. 17–32 (2010). doi:10.4230/LIPIcs.STACS.2010.2495 Google Scholar
- 38.Niedermeier, R., Rossmanith, P.: On efficient fixed-parameter algorithms for weighted vertex cover. J. Algorithms 47(2), 63–77 (2003). doi:10.1016/S0196-6774(03)00005-1 MathSciNetMATHCrossRefGoogle Scholar
- 39.Raman, V., Ramanujan, M.S., Saurabh, S.: Paths, flowers and vertex cover. In: Proc. 19th ESA, pp. 382–393 (2011). doi:10.1007/978-3-642-23719-5_33 Google Scholar
- 40.Razgon, I., O’Sullivan, B.: Almost 2-sat is fixed-parameter tractable. J. Comput. Syst. Sci. 75(8), 435–450 (2009). doi:10.1016/j.jcss.2009.04.002 MathSciNetMATHCrossRefGoogle Scholar
- 41.Schrijver, A.: Combinatorial Optimization. Polyhedra and Efficiency. Springer, Berlin (2003) MATHGoogle Scholar
- 42.Soleimanfallah, A., Yeo, A.: A kernel of order 2k−c for vertex cover. Discrete Math. 311(10–11), 892–895 (2011). doi:10.1016/j.disc.2011.02.014 MathSciNetMATHCrossRefGoogle Scholar
- 43.Uhlmann, J., Weller, M.: Two-layer planarization parameterized by feedback edge set. In: Proc. 7th TAMC, pp. 431–442 (2010). doi:10.1007/978-3-642-13562-0_39 Google Scholar
- 44.Yap, C.K.: Some consequences of non-uniform conditions on uniform classes. Theor. Comput. Sci. 26, 287–300 (1983). doi:10.1016/0304-3975(83)90020-8 MathSciNetMATHCrossRefGoogle Scholar
- 45.Zito, J.: The structure and maximum number of maximum independent sets in trees. J. Graph Theory 15(2), 207–221 (1991). doi:10.1007/s00224-012-9393-4 MathSciNetMATHCrossRefGoogle Scholar