Approximation Algorithms for Connected Graph Factors of Minimum Weight
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Abstract
Finding lowcost spanning subgraphs with given degree and connectivity requirements is a fundamental problem in the area of network design. We consider the problem of finding dregular spanning subgraphs (or dfactors) of minimum weight with connectivity requirements. For the case of kedgeconnectedness, we present approximation algorithms that achieve constant approximation ratios for all d≥2⋅⌈k/2⌉. For the case of kvertexconnectedness, we achieve constant approximation ratios for d≥2k−1. Our algorithms also work for arbitrary degree sequences if the minimum degree is at least 2⋅⌈k/2⌉ (for kedgeconnectivity) or 2k−1 (for kvertexconnectivity). To complement our approximation algorithms, we prove that the problem with simple connectivity cannot be approximated better than the traveling salesman problem. In particular, the problem is A P Xhard.
Keywords
Graph factors Edgeconnectivity Vertexconnectivity Approximation algorithms1 Introduction
The traveling salesman problem (MinTSP) is a basic combinatorial optimization problem: given a complete graph G=(V,E) with edge weights that satisfy the triangle inequality, the goal is to find a Hamiltonian cycle of minimum total weight. Phrased differently, we are looking for a subgraph of G of minimum weight that is 2regular, connected, and spanning. While M i n  T S P is N Phard [11, ND22], omitting the requirement that the subgraph must be connected makes the problem polynomialtime solvable [21, 27]. In general, dregular, spanning subgraphs (also called dfactors) of minimum weight can be found in polynomial time using Tutte’s reduction [21, 27] to the matching problem. Cheah and Corneil [2] have shown that deciding whether a given graph G=(V,E) has a dregular connected spanning subgraph is N Pcomplete for every d≥2, where d=2 is just the Hamiltonian cycle problem [11, GT37]. Thus, finding a connected dfactor of minimum weight is also N Phard for all d.
The problem of finding connected dfactors of minimum weight is a fundamental problem in network design, where the usual setting is that there are connectivity and degree requirements. Then the goal is to find a cheap subgraph that meets the connectivity requirements and the degree bounds. Beyond simple connectedness, higher connectivity, such as kvertexconnectivity or kedgeconnectivity, has been considered in order to increase the reliability of the network. Most variants of such problems are N Phard. Because of this, finding good approximation algorithms for such network design problems has been the topic of a significant amount of research [1, 4, 6, 7, 8, 9, 10, 14, 16, 17, 18, 19, 20].
In this paper, we study the problem of finding lowcost spanning subgraphs with given degrees that meet connectivity requirements (they should be kedgeconnected or kvertexconnected for a given k). Violation of the degree constraint is not allowed.
1.1 Problem Definitions and Preliminaries
1.1.1 Graphs and Connectivity
All graphs in this paper are undirected and simple. Let G=(V,E) be a graph. In the following, n=V is the number of vertices.
For a subset \(X \subseteq V\) of vertices, let cut_{ G }(X) be the number of edges in G with one endpoint in X and the other endpoint in \(\overline X = V \setminus X\). For two disjoint sets \(X, Y \subseteq V\) of vertices, let cut_{ G }(X,Y) be the number of edges in G with one endpoint in X and the other endpoint in Y.
Two vertices u,v∈V are locally kedgeconnected in G if there are at least k edgedisjoint paths from u to v in G. Equivalently, u and v are locally kedgeconnected in G if cut_{ G }(X)≥k for all \(X \subseteq V\) with u∈X and v∉X. Local kedgeconnectedness is an equivalence relation as it is symmetric, reflexive, and transitive. A graph G is kedgeconnected if all pairs of vertices are locally kedgeconnected in G.
Let \(X \subseteq V\). We call X a kedgeconnected component of G if the subgraph induced by X is kedgeconnected. We call X a locally kedgeconnected component of G if all u,v∈X are locally kedgeconnected in G. Note that every kedgeconnected component of G is also a locally kedgeconnected component of G, but the reverse is not true.
A graph G is kvertexconnected, if the graph induced by the vertices V∖K is connected for all sets \(K \subseteq V\) with K≤k−1. Equivalently, for any two nonadjacent vertices u,v∈V, there exist at least k vertexdisjoint paths connecting u and v in G.
We note that testing if two vertices are locally kedgeconnected, if a graph is kedgeconnected, or if a graph is kvertexconnected can be done in polynomial time. For an overview of connectivity and algorithms for computing connectivity and connected components, we refer to two surveys [13, 15].
For a vertex v∈V, let N _{ G }(v) = N(v)={u∈V∣{u,v}∈E} be the neighbors of v in G. The graph G is dregular if N(v) = d for all v∈V. A dregular spanning subgraph of a graph is called a dfactor. Instead of using the same value d for all vertices, we also consider spanning subgraphs where the degree of each vertex v is required to be d _{ v } (Section 4).
By abusing notation, we identify a set \(X \subseteq V\) of vertices with the subgraph induced by X. Similarly, if the set of vertices is clear from the context, we identify a set F of edges with the graph (V,F).
1.1.2 Problem Definitions
Let G=(V,E) be an undirected, complete graph with nonnegative edge weights w. The edge weights are assumed to satisfy the triangle inequality, i.e., w({u,v})≤w({u,x}) + w({x,v}) for all distinct u,v,x∈V. For some set \(F \subseteq E\) of edges, we denote by \(w(F) = {\sum }_{e \in F} w(e)\) the sum of its edge weights. The weight of a subgraph is the weight of its edge set.
The problems considered in this paper are the following: as input, we are given G and w as above. Then M i nd R e gk E d g e denotes the problem of finding a kedgeconnected dfactor of G of minimum weight. Similarly, M i nd R e gk V e r t e x denotes the problem of finding a kvertexconnected dfactor of G of minimum weight.

The two problems M i nd R e g1E d g e and M i nd R e g1V e r t e x are identical for all d since 1edgeconnectedness and 1vertexconnectedness are simply connectedness.

For k∈{1,2}, the problems M i n2R e gk E d g e and M i n2R e gk V e r t e x are identical to the traveling salesman problem (MinTSP).

For even d and k, the problems M i nd R e g(k−1)E d g e and M i nd R e gk E d g e are identical. For even d, every dfactor can be decomposed into d/2 2factors. Thus, the size of every cut is even. Therefore, every dregular (k−1)edgeconnected graph is automatically kedge connected for even k.

For k∈{1,2,3}, the two problems M i n3R e gk E d g e and M i n3R e gk V e r t e x are identical since edge and vertexconnectivity are equal in cubic graphs [28, Theorem 4.1.11].
We also consider the generalizations of M i nd R e gk E d g e and M i nd R e gk V e r t e x to arbitrary degree sequences: for M i nd G e nk E d g e, we are given as additional input a degree requirement \(d_{v} \in \mathbb {N}\) for every vertex v. The parameter d is a lower bound for the degree requirements, i.e., we have d _{ v }≥d for all vertices v. The goal is to compute a kedgeconnected spanning subgraph in which every vertex v is adjacent to exactly d _{ v } vertices. Mind Genk Vertex is analogously defined for kvertexconnectivity. For the sake of readability, we restrict the presentation of our algorithms in Sections 2 and 3 to M i nd R e gk V e r t e x and M i nd R e gk E d g e, and we state the generalized results for M i nd G e nk V e r t e x and M i nd G e nk E d g e only in Section 4.
We use the following notation: OptE^{ k } denotes a kedgeconnected subgraph of minimum weight. OptV^{ k } denotes a kvertexconnected subgraph of minimum weight. For both, no degree requirements have to be satisfied. OptF_{ d } denotes a (not necessarily connected) dfactor of minimum weight. \(\text {optEF}_{d}^{k}\) and \(\text {OptVF}_{d}^{k}\) denote minimumweight kedgeconnected and kvertexconnected dfactors, respectively.
We have \(w(\text {OptF}_{d}) \leq w(\text {optEF}_{d}^{k}) \leq w(\text {OptVF}_{d}^{k})\) since every kvertexconnected graph is also kedgeconnected. Both \(w(\text {optEF}_{d}^{k})\) and \(w(\text {OptVF}_{d}^{k})\) are monotonically increasing in k. Furthermore, \(w(\text {OptE}^{k}) \leq w(\text {optEF}_{d}^{k})\) for every d and \(w(\text {OptV}^{k}) \leq w(\text {OptVF}_{d}^{k})\) for every d.
We denote by MST a minimumweight spanning tree of G.
1.2 Previous and Related Results
Without the triangle inequality, the problem of computing minimum weight kvertexconnected spanning subgraphs can be approximated within a factor of \(O(\log k)\) [3], and the problem of computing minimum weight kedgeconnected spanning subgraphs can be approximated with in a factor of 2 [16]. However, no approximation at all seems to be possible without the triangle inequality if we ask for specific degrees. This follows from the inapproximability of nonmetric TSP [29, Section 2.4].
With the triangle inequality, we obtain the same factor of 2 for kedgeconnected subgraphs of minimum weight without degree requirements [16]. For kvertexconnected spanning subgraphs of minimum weight without degree constraints, Kortsarz and Nutov [17, Theorem 4.2] gave a \(\bigl (2 + \frac {k1}n\bigr )\)approximation algorithm.
Overview of approximation ratios for M i nd R e gk V e r t e x
k  d  ratio  reference 

∈{1,2}  =2  1.5  same as problem as M i nT S P [29, Section 2.4] 
∈{1,2,3}  =3  same as M i nd R e gk E d g e  
=1  arbitrary  same as M i nd R e g1E d g e  
≥2  = k  \(2+\frac {k1}n + \frac 1k\)  Chan et al. [1] 
≥2  =2k−1  \(5 + \frac {2k2}n + \frac 2k\)  Theorem 2.2 
≥2  ≥2k  \(5 + \frac {2k2}n\)  Corollary 2.3 
Fukunaga and Nagamochi [8] considered the problem of finding a minimumweight kedgeconnected spanning subgraph with given degree requirements. Different from the problem that we consider, they allow multiple edges between vertices. This considerably simplifies the problem as one does not have to take care to avoid multiple edges when constructing the approximate solution. For this relaxed variant of the problem, they obtain approximation ratios of 2.5 for even k and \(2.5 + \frac {1.5}k\) for odd k if the minimum degree requirement is at least 2. We remark that, although an optimal solution with multiple edges cannot be heavier than an optimal solution without multiple edges, an approximation algorithm for the variant with multiple edges does not imply an approximation algorithm for the variant without multiple edges and vice versa.
In many cases of algorithms for network design with degree constraints, only bounds on the degrees are given or some violation of the degree requirements is allowed to simplify the problem. Fekete et al. [7] devised an approximation algorithm for the boundeddegree spanning tree problem. Given lower and upper bounds for the degree of every vertex, spanning trees can be computed that violate every degree constraint by at most 1 and whose weight is no more than the weight of an optimal solution [26]. Often, network design problems are considered as bicriteria problems, where the goal is to simultaneously minimize the total costs and the violation of the degree requirements [9, 10, 18, 19, 20]. In contrast, our goal is to meet the degree requirements exactly.
Recently [23], M i nd R e g1E d g e has been considered for the case that d grows with the number n of vertices. It turns out that the problem becomes simpler for large enough d, admitting a PTAS for d≥n/c for any constant c.
1.3 Our Contribution
We devise polynomialtime approximation algorithms for M i nd R e gk V e r t e x (Section 2) and for M i nd R e gk E d g e (Section 3). Our algorithms can be generalized to arbitrary degree sequences, as long as the minimum degree requirement is at least 2k−1 for vertex connectivity or at least 2⌈k/2⌉ for edge connectivity (Section 4).
Roughly, we obtain an approximation ratio of about 5 for M i nd R e gk V e r t e x for d≥2k−1, an approximation ratio of roughly 4 for M i nd R e gk E d g e for odd d≥k+1 and a ratio of 2.5 for M i nd R e gk E d g e for even d≥k. The precise approximation ratios are summarized in Tables 1 and 2.
As far as we are aware, there do not exist any approximation results for the problem of finding subgraphs with exact degree requirements. The only exception that we are aware of is the work by Fukunaga and Nagamochi [8]. However, they allow multiple edges in their solutions, which seems to make the problem simpler to approximate.
The highlevel ideas of our algorithms are as follows. For vertexconnectivity, the idea is to compute a kvertexconnected kregular graph and a (possibly not connected) dfactor. We iteratively add edges from the kvertexconnected graph to the dfactor while maintaining the degrees until we obtain a kvertexconnected dfactor. This works for d≥2k−1 (Lemma 2.1).
For edgeconnectivity, our initial idea was to iteratively increase the connectivity from k−1 to k by considering the kedgeconnected components of the current solution and adding edges carefully. However, this does not work as kedgeconnected components are not guaranteed to exist in (k−1)edgeconnected graphs. Instead, we introduce kspecial components (Definition 3.2). By connecting the kspecial components carefully, we can increase the edgeconnectivity of the graph (Lemma 3.11). Every increase of the edgeconnectivity costs at most a fraction O(1/k) of the weight of the optimal solution (Lemma 3.18), yielding constant factor approximations for all k.
Finally, we prove that M i nd R e g1E d g e is A P Xhard. We extend this result and the N Phardness of finding connected dfactors to the case where d grows with the number n of vertices.
2 Vertex Connectivity
In this section, we consider M i nd R e gk V e r t e x for d≥2k−1. The basis of the algorithm (Algorithm 1) is the following: Assume that we have a kvertex connected kfactor H and a dfactor F that lacks kvertexconnectedness. Then we iteratively add edges from H to F to make F kvertexconnected as well. More precisely, we try to add an edge e∈H∖F to increase the connectivity of F. To maintain that F is dregular, we have to add another edge and remove two edges of F. If, in the course of this process, we never have to remove an edge of H from F, then the algorithm terminates with a kvertexconnected dregular graph.
With this initialization, we iteratively add edges from H to F while maintaining dregularity of F. This works as long as d is sufficiently large according to the following lemma. We parametrize the maximum degree of H by ℓ in order to be able to get a slight improvement for larger d (Corollary 2.3).
Lemma 2.1
Let k,ℓ≥2 and d≥k+ℓ−1. Let G=(V,E) be an undirected complete graph. Let F be a dfactor of G that is not kvertex connected, and let H be a kvertex connected subgraph of G that has a maximum degree of at most ℓ.
 1.
The vertices u _{1} and u _{2} are not connected via k vertexdisjoint paths in F.
 2.
{u _{1} ,v _{1} },{u _{2} ,v _{2}}∈F∖H.
 3.
{v _{1} ,v _{2}}∉F.
Proof 1
Since F is not kvertexconnected, there exists a subset \(X \subseteq V\) of vertices with X≤k−1 such that the subgraph \(\tilde G_{F}\) of F induced by V∖X is not connected. Since H is kvertexconnected, the subgraph \(\tilde G_{H}\) of H induced by V∖X is connected. This means that there exists an edge e={u _{1},u _{2}} in \(\tilde G_{H}\) that connects two different components of \(\tilde G_{F}\). In particular, e∈H∖F and u _{1} and u _{2} are not connected via k vertexdisjoint paths in F.
Let \({e_{1}^{1}}, \ldots , {e_{d}^{1}}\) be the edges of F incident to u _{1}. Since H has a maximum degree of at most ℓ and the edge e is incident to u _{1}, at most ℓ−1 of these d edges are contained in H. Because of this and since d≥k + ℓ−1, at least k of these edges are not contained in H. Let \({e_{1}^{1}}, \ldots , {e_{k}^{1}}\) be k such edges.
In the same way, let \({e_{1}^{2}}, \ldots , {e_{k}^{2}}\) be k edges of F∖H incident to u _{2}.
Let \({e_{i}^{j}} = \{u_{j}, z_{j,i}\}\). We call z _{1,i } and \(z_{2, i^{\prime }}\) connected if \(z_{1,i} = z_{2, i^{\prime }}\) or if they are connected with an edge in F. If all endpoints z _{1,1},…,z _{1,k } were connected to all endpoints z _{2,1},…,z _{2,k }, then this would give us k vertexdisjoint paths from u _{1} to u _{2}, contradicting the assumption of the lemma. Thus, there exist \({e_{i}^{1}} = \{u_{1}, z_{1, i}\}\) and \(e_{i^{\prime }}^{2} = \{u_{2}, z_{2, i^{\prime }}\}\) such that z _{1,i } and \(z_{2, i^{\prime }}\) are not connected. We set v _{1} = z _{1,i } and \(v_{2} = z_{2, i^{\prime }}\). These vertices have the desired properties. □
With this lemma, we can prove the main result of this section.
Theorem 2.2
For \(k, d \in \mathbb {N}\) with k≥2 and d≥2k−1, Algorithm 1 is a polynomialtime approximation algorithm for M i ndR e gkV e r t e x with an approximation ratio of \(5 + \frac {2k2}n + \frac 2k\).
Proof 2
Because of Lemma 2.1 with ℓ = k, we can always find vertices v _{1} and v _{2} as required in line 6, and such vertices can be found in polynomial time. Since no edge of H is ever removed from F and every iteration adds one or two edges of H to F, the while loop runs through at most H iterations. Thus, the overall runningtime is bounded by a polynomial.
Let us analyze the approximation ratio. If we add an edge e∈H to F, then we add in fact e={u _{1},u _{2}} and \(e^{\prime } = \{v_{1}, v_{2}\}\). On the other hand, we remove {u _{1},v _{1}} and {u _{2},v _{2}} from F. By the triangle inequality, \(w(e^{\prime }) \leq w(e) + w(\{u_{1}, v_{1}\}) + w(\{u_{2}, v_{2}\})\). Thus, if we add e and \(e^{\prime }\), then the weight of F increases by at most 2w(e). Therefore, the total weight of R is bounded by w(R)≤w(OptF_{ d })+2w(H).
Algorithm 1 also works for k=1, but for this case, there already exist better approximation algorithms (see Table 1).
With the slightly stronger assumption d≥2k, we can get a slightly better approximation ratio.
Corollary 2.3
For \(k, d \in \mathbb {N}\) with k≥2 and d≥2k, there exists a polynomialtime approximation algorithm for M i ndR e gkV e r t e x with an approximation ratio of \(5 + \frac {2k2}n\).
Proof 3
In line 3 of Algorithm 1, we compute a kvertexconnected graph H of maximum degree k+1 instead of a kregular kvertexconnected graph. According to Chan et al. [1], this can be done in polynomial time with w(H)≤w(K). Now we use Lemma 2.1 with ℓ = k+1. □
3 EdgeConnectivity
In this section, we present an approximation algorithm for M i nd R e gk E d g e for all combinations of d and k, provided that d≥2⌈k/2⌉. This means that the algorithm works for all d≥k with the only exception being the case of odd d = k. It includes the case of simple connectivity, i.e., the case of k=1.
The main idea of our algorithm is as follows: We start by computing a dfactor (without requiring any connectedness). Then we iteratively increase the connectivity as follows: First, we identify edges that we can safely remove without decreasing the connectivity. Second, we find edges that we can add in order to increase the connectivity while repairing the dregularity that we have destroyed in the first step.
One might be tempted to use the kedgeconnected components of the dfactor in order to increase the edgeconnectivity from k−1 to k. This works for k=1 and k=2. However, for larger k, the catch is that there need not be enough kedgeconnected components, and it is in fact possible to find (k−1)edgeconnected graphs that are dregular with d≥k that do not contain any nontrivial kedgeconnected component. To circumvent this problem, we introduce the notion of kspecial components, which have the desired properties.
3.1 GraphTheoretic Preparation
Different from the rest of the paper, the graph G=(V,E) is not necessarily complete in this section.
Lemma 3.1

X≥k+1.

cut _{ G } (X)≥k.
Proof 4
Let ℓ=X. Every vertex in X can be adjacent to at most X−1 = ℓ−1 other vertices of X. Since G has a minimum degree of at least k, every vertex of X must have at least k−ℓ+1 neighbors outside of X. This shows cut_{ G }(X)≥(k−ℓ+1)⋅ℓ. For ℓ=1 and ℓ = k, this last expression evaluates to k. Since it is a concave function of ℓ, we have cut_{ G }(X)≥k for 1≤ℓ≤k. □
The following definition of kspecial components is crucial for the whole Section 3. As far as we are aware, this definition has not appeared yet in the literature.
Definition 3.2
Let \(k \in \mathbb {N}\), and let G=(V,E) be a graph. We call \(L \subseteq V\) a kspecial component in G if cut_{ G }(L)≤k−1 and L is locally kedge connected in G.
For k=1, the kspecial components are the connected components of G. For k=2, the kspecial components are the leaves of the following graph: we have a node for every 2edgeconnected component. Two nodes of this graph are connected if the corresponding 2edgeconnected components are connected via a single edge. This graph is a tree. The 2special components of G correspond to the leaves of this tree.
Let us collect some facts about kspecial components.
Lemma 3.3
Let G have a minimum degree of at least k, and let L be a kspecial component in G. Then L≥k+1.
Proof 5
This follows immediately from Lemma 3.1 and Definition 3.2. □
Lemma 3.4
Let G be a graph. If L is a kspecial component, then L is a maximal locally kedgeconnected component. If L and \(L^{\prime }\) are kspecial, then either \(L = L^{\prime }\) or \(L \cap L^{\prime } = \emptyset \).
Proof 6
If L were not maximal, then we would have cut_{ G }(L)≥k. If L and \(L^{\prime }\) intersect but are not identical, then cut_{ G }(L)≥k or \(\text {cut}_{G}(L^{\prime }) \geq k\). Hence, if L and \(L^{\prime }\) intersect, then they must be identical. □
The following lemma is crucial as it proves the existence of kspecial components.
Lemma 3.5
Let k≥1. Let G=(V,E) be a (k−1)edgeconnected graph. Then every nonempty vertex set \(X \subsetneq V\) either contains a kspecial component or satisfies cut _{ G } (X)≥k.
Proof 7
Assume to the contrary that there exists a set X with cut_{ G }(X)≤k−1 that does not contain a kspecial component. We choose X minimal in the sense such that no nonempty proper subset \(Y \subsetneq X\) with cut_{ G }(Y)≤k−1 does not contain a kspecial component.
Now consider any \(Y \subsetneq X\). If Y contains a kspecial component \(L \subseteq Y\), then also X contains a kspecial component. We can conclude that cut_{ G }(Y)≥k for all nonempty \(Y \subsetneq X\).
If X itself is locally kedgeconnected, then X is a kspecial component because cut_{ G }(X)≤k−1 by assumption. Thus, we can conclude that X is not locally kedgeconnected. Hence, there exist vertices u,v∈X and a set \(U \subseteq V\) with u∈U and \(v \in V \setminus U = \overline U\) such that cut_{ G }(U)≤k−1.
We consider three cases. The first case is that \(\overline X \cap U = \emptyset \). We only have to deal with k _{2}, k _{3}, and k _{5}. Among others, we have the constraint k _{3} + k _{5}≥k since \(X \cap U\) is a proper subset of X, which implies \(\text {cut}_{G}(X \cap U) \geq k\). And we have the constraint k _{3} + k _{5}≤k−1 since cut_{ G }(U)≤k−1 and \(\overline X \cap U = \emptyset \). These two cannot be satisfied simultaneously.
The second case is that \(\overline X \cap \overline U = \emptyset \). In the same way as in Case 2, we have k _{3} + k _{6}≥k and k _{3} + k _{6}≤k−1, which cannot be satisfied simultaneously.
The last and general case is that both \(\overline X \cap U \neq \emptyset \) and \(\overline X \cap \overline U \neq \emptyset \). Since \(X \cap U\) and \(X \cap \overline U\) are proper subsets of X, we have \(\text {cut}_{G}(X \cap U), \text {cut}_{G}(X \cap \overline U) \geq k\). This translates to k _{1} + k _{3} + k _{5}≥k and k _{2} + k _{3} + k _{6}≥k.
We have cut_{ G }(X)≤k−1, which implies k _{1} + k _{2} + k _{5} + k _{6}≤k−1. And we have cut_{ G }(U)≤k−1, we implies k _{3} + k _{4} + k _{5} + k _{6}≤k−1.
Finally, the graph G is (k−1)edgeconnected. Thus, there are at least k−1 edgedisjoint paths from \(\overline X \cap U\) to \(\overline X \cap \overline U\). This translates to \(k_{4} + k_{5} + k_{6} + \min (k_{1}, k_{2}, k_{3}) \geq k1\). The latter corresponds to three inequalities: k _{ i } + k _{4} + k _{5} + k _{6}≥k−1 for i∈{1,2,3}.
The purpose of the next few lemmas is to show that we can always remove an edge from a kspecial component without decreasing the connectedness of the whole graph. In the following, let m=⌈k/2⌉+1. It turns out that the graph induced by a kspecial component contains a locally medgeconnected component (Lemma 3.7). The next lemma is useful for this.
Lemma 3.6
Let k≥1, let G=(V,E) be a (k−1)edgeconnected graph, and let \(L \subseteq V\) be a kspecial component of G. Then L is (⌊k/2⌋+1)edgeconnected.
Proof 8
Consider u,v∈L. As u is locally kedgeconnected to v, there are at least k edgedisjoint paths from u to v. Since L is kspecial, we have cut_{ G }(L)≤k−1. Thus, at most \(\lfloor \frac {k1}2\rfloor \) of these edgedisjoint paths can leave L. Hence, there are at least \(k  \lfloor \frac {k1}2\rfloor = \lfloor k/2 \rfloor +1\) paths running solely through vertices in L. □
Lemma 3.7
Let k≥1, and let m=⌈k/2⌉+1. Let G=(V,E) be a (k−1)edgeconnected graph of minimum degree at least 2⌈k/2⌉, and let L be a kspecial component of G. Then there exists an \(X \subseteq L\) such that X is a locally medgeconnected component in L and X≥k+1.
Proof 9
For even k, Lemma 3.6 implies that we can choose X = L. We have X≥k+1 by Lemma 3.3.
For k=1, we have m=2, and G has a minimum degree of 2. The kspecial components are just the connected components of G. Every connected component in a graph of minimum degree 2 contains a 2edgeconnected component, which satisfies X≥2.
Now let k≥3 be odd. The minimum degree in this case is 2⌈k/2⌉ = k+1. By Lemma 3.6, we know that L is (m−1)edgeconnected. If L is also medgeconnected, then we can choose X = L. We have X≥k+1 by Lemma 3.3.
Otherwise, there exists a set \(Y \subseteq L\) with cut_{ L }(Y)≤m−1. Lemma 3.5 implies the existence of an mspecial component \(X \subseteq Y\) of L. This implies that X is locally medgeconnected in L. To finish the proof, we have to show that ℓ=X≥k+1. We have cut_{ L }(X)≤m−1 since X is mspecial within L.
We have cut_{ G }(L) = a + b≤k−1 since L is a kspecial component in G. We have cut_{ L }(X) = c≤m−1 since X is an mspecial component in L. We have cut_{ G }(X) = a + c≥k+1 because x has a degree of at least k+1. The first two inequalities imply a + b + c≤k + m−2. From this and the third one, we obtain b≤m−3. Choose any v∈L∖X. Since L is a kspecial component in G, we have at least k edgedisjoint paths from x to v in G. At most c≤m−1 of these paths can run solely through L. Thus, b≥k−c≥k−m+1. Together with the previously derived b≤m−3, we obtain k−m+1≤m−3. This is equivalent to k+4≤2m, which does not hold since \(m = \lceil k/2 \rceil +1 = \frac {k+3}2\). We can conclude that X=1 is impossible. □
Given Lemma 3.7, the next lemma follows. The edges {u _{ i },v _{ i }} mentioned in this lemma are the edges that we can safely remove. The resulting graph will remain (k−1)edgeconnected (Lemma 3.10). The vertices u _{ i } and v _{ i } in the next lemma will be chosen from \(X_{i} \subseteq L_{i}\), where X _{ i } is a locally medgeconnected component in L _{ i } as in Lemma 3.7.
Lemma 3.8

{u _{ i } ,v _{ i } }∈E for all i.

{u _{ i } ,v _{ j } }∉E for all i≠j.

There exist at least m edgedisjoint paths from u _{ i } to v _{ i } in the graph induced by L _{ i } for every i.
Proof 10
Consider any i∈{1,…,s}. According to Lemma 3.7, there exists a locally medgeconnected set \(X_{i} \subseteq L_{i}\) with X _{ i }≥k+1. Since X _{ i }≥k+1 and cut_{ G }(L _{ i })≤k−1 (because L _{ i } is kspecial), there must be a vertex u _{ i }∈X _{ i } with \(N(u_{i}) \subseteq L_{i}\).
We choose v _{ i } to be another vertex in the locally medgeconnected component X _{ i } contained in L _{ i } such that {u _{ i },v _{ i }}∈E. If such a v _{ i } exists, then there are m edgedisjoint paths from u _{ i } to v _{ i } in L _{ i } since u _{ i },v _{ i }∈X _{ i } and X _{ i } is locally medgeconnected in L _{ i }.
Such vertices v _{ i } exist because of the construction of the sets X _{ i } in Lemma 3.7: Either X _{ i } = L _{ i }. Then this follows since \(N(u_{i}) \subseteq L_{i} = X_{i}\). Or \(X_{i} \subsetneq L_{i}\). In this case, we have \(\text {cut}_{L_{i}}(X_{i}) \leq m1 = \lceil k/2\rceil \) since X _{ i } is an mspecial component in L _{ i }. Since u _{ i } has at least 2⌈k/2⌉ neighbors, all of which are in L _{ i }, at least one v _{ i }∈X _{ i } that is adjacent to u _{ i } must exist. □
Lemma 3.9
Let G=(V,E) be a (k−1)edgeconnected graph of minimum degree at least 2⌈k/2⌉ with kspecial components L _{1} ,…,L _{ s } . Let u _{1} ,…,u _{ s } and v _{1} ,…,v _{ s } be chosen as in Lemma 3.8. Let \(Q = \bigl \{\{u_{i}, v_{i}\} \mid 1 \leq i \leq s\bigr \}\).
For all i and j with i≠j, there does not exist a set \(C \subseteq E\) with C≤k−1 such that u _{ i } is disconnected from v _{ i } and u _{ j } is disconnected from v _{ j } in G−Q−C.
Proof 11
Within L _{ i }, there are m edgedisjoint paths from u _{ i } to v _{ i } by Lemma 3.8. After removing Q from G, there remain at least m−1=⌈k/2⌉ edgedisjoint paths within L _{ i }.
The same holds for L _{ j } with u _{ j } and v _{ j }. Thus, any set C of edges that simultaneously disconnects u _{ i } from v _{ i } and u _{ j } from v _{ j } must satisfy C≥2⋅⌈k/2⌉≥k. □
Lemma 3.10
Let G=(V,E) be a (k−1)edgeconnected graph of minimum degree at least 2⌈k/2⌉ with kspecial components L _{1} ,…,L _{ s } , and let u _{1} ,…,u _{ s } and v _{1} ,…,v _{ s } be chosen as in Lemma 3.8. Let \(Q = \bigl \{\{u_{i}, v_{i}\} \mid 1 \leq i \leq s\bigr \}\) . Then G−Q is (k−1)edgeconnected.
Proof 12
We have to show that cut_{ G−Q }(X)≥k−1 for all nonempty \(X \subsetneq V\). To do this, we distinguish two cases: In the first case, for every i, either u _{ i },v _{ i }∈X or u _{ i },v _{ i }∉X. In this case, cut_{ G−Q }(X)=cut_{ G }(X)≥k−1 since no edges between X and V∖X are removed.
In the second case, there exists an i such that \(\{u_{i}, v_{i}\} \cap X = 1\). If there are two or more i with this property, then cut_{ G−Q }(X)≥k−1 follows from Lemma 3.9.
Now assume that there is only a single i with \(\{u_{i}, v_{i}\} \cap X = 1\). Without loss of generality, we assume that u _{ i }∈X and v _{ i }∉X. Since u _{ i } and v _{ i } are locally kedgeconnected in G, we have cut_{ G }(X)≥k. Only the edge {u _{ i },v _{ i }} is removed from this cut, thus cut_{ G−Q }(X)≥k−1. □
By removing the edges {u _{ i },v _{ i }}∈Q and adding the edges {u _{ i },v _{ i+1}}∈S, we construct a kedgeconnected graph from the (k−1)edgeconnected graph G according to the following lemma.
Lemma 3.11
Let G=(V,E) be a (k−1)edgeconnected graph of minimum degree at least 2⌈k/2⌉ with kspecial components L _{1} ,…,L _{ s } , and let u _{1} ,…,u _{ s } and v _{1} ,…,v _{ s } be chosen as in Lemma 3.8. Let \(Q = \bigl \{\{u_{i}, v_{i}\} \mid 1 \leq i \leq s\bigr \}\) , and let \(S = \bigl \{\{u_{i}, v_{i+1}\} \mid 1 \leq i \leq s\bigr \}\) , where arithmetic is modulo s.
Then the graph \(\tilde G = G  Q + S\) is kedgeconnected.
Proof 13
We prove the lemma by a series of claims. In the following, arithmetic is modulo s.
Claim 3.12
For every i, the vertices u _{ i } and v _{ i+1 } are locally kedgeconnected in \(\tilde G\).
Proof 14
The graph G−Q is (k−1)edgeconnected and S contains the edge {u _{ i },v _{ i+1}}. □
Claim 3.13
For all i and j, the vertices u _{ i } and u _{ j } are locally kedgeconnected in \(\tilde G\).
Proof 15
Without loss of generality, we assume i<j.
Assume to the contrary that there is a set C of at most k−1 edges such that C disconnects u _{ i } from u _{ j }. Since G−Q is (k−1)edgeconnected by Lemma 3.10, we must have \(C \cap S = \emptyset \).
Lemma 3.9 says that if C disconnects u _{ a } from v _{ a } for some a, then it cannot disconnect u _{ b } from v _{ b } for a≠b. Thus, C can disconnect at most one pair u _{ a }, v _{ a }. Consider the paths from u _{ i } to u _{ j } via v _{ i+1},u _{ i+1},v _{ i+2},u _{ i+2},…,v _{ j−1},u _{ j−1},v _{ j } or via v _{ i },u _{ i−1},v _{ i−1},u _{ i−2},…,v _{ j+2},u _{ j+1},v _{ j+1} after adding S. The set S contains all the connections between u and v vertices in these paths. Since C can disconnect at most one pair u _{ a },v _{ b } by Lemma 3.9, one of these paths must still exist. Thus, u _{ i } and v _{ i } are still connected in \(\tilde G\) after removing at most k−1 edges. □
Claim 3.14
For all i and j, vertices u _{ i } , v _{ i } , u _{ j } , and v _{ j } are pairwise locally kedgeconnected in \(\tilde G\).
Proof 16
This follows by transitivity of local kedgeconnectedness and Claims 3.12 and 3.13 above. □
Claim 3.15
For all i, the vertices in L _{ i } are locally kedgeconnected in \(\tilde G\).
Proof 17
We show that every x∈L _{ i } is locally kedgeconnected to u _{ i } in \(\tilde G\). Then the claim follows by transitivity of local kedgeconnectedness.
Let \(X \subsetneq V\) with x∈X and u _{ i }∉X. If v _{ j }∈X (including the case j = i) or u _{ j }∈X for some j≠i, then \(\text {cut}_{\tilde G}(X) \geq k\) by Claim 3.14. Thus, the case u _{1},…,u _{ s },v _{1},…,v _{ s }∉X remains to be considered. Since L _{ i } is locally kedgeconnected in G, we have cut_{ G }(X)≥k. This cut does not change by removing Q. Hence, \(\text {cut}_{\tilde G}(X) \geq k\). □
Transitivity of local kedgeconnectedness, Claim 3.14, and Claim 3.15 imply that \(L = \bigcup _{i=1}^{s} L_{i}\) is locally kedgeconnected in \(\tilde G\). The vertices that are not part of any kspecial component in G remain to be considered.
Consider an arbitrary nonempty \(X \subsetneq V\). If we can show that \(\text {cut}_{X}(\tilde G) \geq k\), then we have completed the proof of this lemma. If \(X \cap L \neq \emptyset \) and \((V \setminus X) \cap L \neq \emptyset \), then \(\text {cut}_{\tilde G}(X) \geq k\) because L is locally kedgeconnected in \(\tilde G\). Thus, \(L \subseteq X\) or \(L \subseteq V \setminus X\). By symmetry, we restrict ourselves to the second case. Then we have \(\text {cut}_{G}(X) = \text {cut}_{\tilde G}(X)\) because no edge connecting X to V∖X is in Q or S.
If cut_{ G }(X)≤k−1, then X contains a kspecial component by Lemma 3.5, contradicting our assumption. Thus, we have cut_{ G }(X)≥k. □
To conclude this section, we remark that the kspecial components of a graph can be found in polynomialtime: local kedgeconnectedness can be tested in polynomial time. Thus, we can find locally kedgeconnected components in polynomial time. Since kspecial components are maximal locally kedgeconnected components, we just have to compute a partition of the graph into locally kedgeconnected components and check whether less than k edges leave such a component. Therefore, the sets L _{ i } and \(X_{i} \subseteq L_{i}\) as well as the vertices u _{ i } and v _{ i } with the properties as in Lemmas 3.7 and 3.8 can be computed in polynomial time.
Finally, we need the following lemma for improving the approximation ratios in the next section. It basically implies that we can jump from disconnected graphs directly to 2edgeconnected graphs in one iteration.
Lemma 3.16
 1.
There exists a set of edges \(Q = \bigl \{\{u_{i}, v_{i}\} \mid 1 \leq i \leq s\bigr \}\) such that {u _{ i } ,v _{ i } }∈E, u _{ i } ,v _{ i } ∈L _{ i } , and neither u _{ i } nor v _{ i } are connected to any vertex outside L _{ i }.
 2.
The graph \(G^{\prime } = G  Q\) contains exactly the same connected components as G.
 3.
Let \(S = \bigl \{\{u_{i}, v_{i+1}\} \mid 1 \leq i \leq s\bigr \}\) , where arithmetic is modulo s. Then the graph \(\tilde G = G  Q + S\) is 2edgeconnected.
Proof 18
Every 2special component is 2edgeconnected. Consider any 2special component L _{ i } of G. We have cut_{ G }(L _{ i })≤1 by definition. If cut_{ G }(L _{ i })=0, then L _{ i } contains at least three vertices since G has minimum degree 2. Hence, the existence of an edge {u _{ i },v _{ i }}∈E with u _{ i },v _{ i }∈L _{ i } follows. If cut_{ G }(L _{ i })=1, then there is one vertex x _{ i } that connects L _{ i } to the rest of the graph. This vertex must be incident to at least two other vertices of L _{ i } as L _{ i } is 2edgeconnected. Now a similar argument as for the case cut_{ G }(L _{ i })=0 applies. This proves Item (1).
Since every 2special component is 2edgeconnected, removal of Q does not cause any 2special component to split. Thus, we have Item (2).
Let us now prove Item (3). The graph \(\tilde G\) is connected: any two connected components of G, which are still connected in \(G^{\prime }\), contain 2special components L _{ i } and L _{ j }, respectively. These are connected via a path that passes through u _{ i },v _{ i+1},u _{ i+1},…,u _{ j−1},v _{ j }. We have to show that \(\tilde G\) remains connected after the removal of any edge.
First, if we remove an edge {u _{ i },v _{ i+1}}∈S, then we still have a path from u _{ i } to v _{ i+1} via v _{ i },u _{ i−1},v _{ i−1},…,v _{ i+2},u _{ i+1}. Second, if we remove a bridge edge e={x,y} of G (a bridge edge of a graph is an edge whose removal increases the number of connected components), then there still must exist a path from x to some u _{ i } that does not visit y. Similarly, there must exist a path from y to some u _{ j } that does not visit x. Now u _{ i } and u _{ j } are connected as in the first case. Third, if we remove any other edge e={x,y} of \(\tilde G\), then x and y belong to the same 2edgeconnected component C of G. If C = L _{ i } is 2special, then there is a path from x to y in G−e. If this path uses {u _{ i },v _{ i }}, we can reroute it via v _{ i+1},u _{ i+1},…,v _{ i−1},u _{ i−1} to obtain a path in \(\tilde G\). If this path does not use {u _{ i },v _{ i }}, then it still exists in \(\tilde G\). If C is not equal to a 2special component, then C is a 2edgeconnected component of \(\tilde G\) and, thus, remains connected after removal of e. □
3.2 Algorithm and Analysis
We analyze correctness and approximation ratio using a series of lemmas.
Lemma 3.17
Let k≥1 be arbitrary, and let p∈{0,2,3,4,…,k−1,k}. Let F _{ p } be computed by Algorithm 2. Then F _{ p } is dregular and pedgeconnected.
Proof 19
By construction of F _{0} and F _{2},…,F _{ k }, all these graphs are dregular. The graph F _{2} is 2edgeconnected by Lemma 3.16 or F _{0} was already 2edgeconnected. Now assume that the lemma holds for some p−1 for p≥3. We know that F _{ p−1} is (p−1)edgeconnected and dregular by induction hypothesis.
In our approximation algorithm, we use Christofides’ algorithm [29, Section 2.4] to compute TSP tours. In order to analyze the approximation ratio and to achieve a constant approximation for all k, we exploit a result that Fukunaga and Nagamochi [8] attributed to Goemans and Bertsimas [12] and Wolsey [30]. It relates the weight of the tour computed by Christofides’ algorithm to the objective value of the relaxation of the integer linear program for kedgeconnected graphs of minimum weight. Fukunaga and Nagamochi state the following result only for the case k=2. We obtain the result below that we need for our purposes by scaling the righthand side of their linear program by a factor of k/2.
Lemma 3.18 (Fukunaga, Nagamochi [8, Theorem 2])
Let T be the TSP tour obtained from Christofides’ algorithm. Then \(w(T) \leq \frac 3k \cdot w(\text {OptE}^{k})\).
We also need the following lemma. (Otherwise, we have to replace Christofides’ algorithm by the spanning tree heuristic for the case of k=1.)
Lemma 3.19
Let T be the TSP tour obtained from Christofides’ algorithm. Then w(T)≤2⋅w(MST).
Proof 20
The weight w(T) can be bounded from above by the sum of an MST plus the weight of a minimumweight perfect matching on the odddegree nodes of this tree. The weight of this minimumweight perfect matching can be bounded from above by the weight of the MST using the triangle inequality. □
Lemma 3.20
Proof 21
Lemma 3.21
In Algorithm 2, we have \(w(F_{2}) \leq w(F_{0}) + \frac 3k \cdot w(\text {optEF}_{d}^{k})\).
Proof 22
The proof is almost identical to the proof of Lemma 3.20. □
Lemma 3.22
Proof 23
The lemma follows by immediately applying Lemma 3.20. □
Theorem 3.23

2.5 for even d,

\(4  \frac 3k\) for odd d and k≥2, and

3 for odd d and k=1.
Proof 24
First, let d be even. We can restrict ourselves to consider even k as discussed in Section 1.1.2. Algorithm 2 calls Algorithm 3 at most k/2−1 times as every call increases the connectivity by at least two. The approximation ratio follows from Lemmas 3.21 and 3.22 and \(w(F_{0}) \leq w(\text {optEF}_{d}^{k})\).
Second, let d be odd and k≥2. We need at most k−2 calls of Algorithm 3 to obtain a kedgeconnected dfactor. Thus, we obtain \(w(F_{k}) \leq \frac {3k6}k \cdot w(\text {optEF}_{d}^{k}) + w(F_{2}) \leq \bigl (4  \frac 3k\bigr ) \cdot w(\text {optEF}_{d}^{k})\) by Lemmas 3.21 and 3.22.
Finally, consider odd d and k=1. In this case, the approximation ratio follows from Lemma 3.19. (Lemma 3.20 yields a worse bound for this case.) □
Algorithm 2 works also for the case of even d = k, but there exists already an approximation algorithm with a ratio of \(2+ \frac 1k\) for this special case [1].
Remark 3.24
We observe that even for M i nd R e g1E d g e for odd d, Algorithm 2 always outputs a 2edgeconnected graph that weighs at most \(3 \cdot w(\text {OptEF}_{d}^{1})\). This limits the approximation ratio of the approximation algorithm as there are instances with \(3 \cdot w(\text {OptEF}_{d}^{1}) = w(\text {OptEF}_{d}^{2})\) (see Proposition 3.5 below). Directly computing a 1edgeconnected solution might result in an improved approximation ratio, but we do not see how to achieve this.
Proposition 3.25
For every odd d≥3, there are instances with \(3 \cdot w(\text {OptEF}_{d}^{1}) = w(\text {OptEF}_{d}^{2})\).
Proof 25
The set of vertices of the instance consists of a vertex v plus d sets V _{1},…,V _{ d } that consist of d+2 vertices each. We set the distance of v to each of the other vertices equal to 1. The distance between each pair of vertices from the same set V _{ i } is 0. Finally, the distance between all pairs of vertices from different sets V _{ i } and V _{ j } is 2. The triangle inequality is satisfied.
An optimal connected dfactor connects v to one vertex of each V _{ i }. Since each V _{ i } has an odd number of vertices and d is odd as well, we can complete the connected dfactor without any further cost. Thus, the total cost is d.
An optimal 2edgeconnected dfactor has a weight of 3d: Because each V _{ i } has an odd number of vertices and d is odd, any 2edgeconnected dfactor must have at least three edges leaving each set V _{ i }. If such an edge e is incident with v, then we charge its weight to V _{ i }. The other possibility is that e is incident with a vertex from some V _{ j }, where j≠i. In this case e has a weight of 2 and we charge a weight of 1 to both V _{ i } and V _{ j }. The total charge of all sets V _{ i } equals the total weight of the 2edgeconnected dfactor. Since each V _{ i } is charged at least 3, the total weight of any 2edgeconnected dfactor is at least 3d. □
4 Generalization to Arbitrary Degree Sequences
Both algorithms of Sections 2 and 3 do not exploit dregularity, but only that the degree of each vertex is at least d. Thus, we immediately get approximation algorithms for M i nd G e nk V e r t e x and M i nd G e nk E d g e, where we have a degree requirement of at least d for each vertex.
For kvertexconnectivity, we require that the minimum degree requirement is at least 2k−1. (For minimum degree at least 2k, we get a small improvement similarly to Corollary 2.3.) For kedgeconnectedness, we require that the minimum degree requirement is at least 2⌈k/2⌉.
Theorem 4.1
For k≥2, Min(2k−1)G e nkV e r t e x can be approximated in polynomial time with an approximation ratio of \(5 + \frac {2k2}n + \frac 2k\).
M i n(2k)G e nkV e r t e x can be approximated in polynomial time with an approximation ratio of \(5 + \frac {2k2}n\).
Theorem 4.2
For k≥2, \(\mathsf {Min}\text {}{(2 \lceil \frac k2 \rceil )}\mathsf {Gen}\text {}k\mathsf {Edge}\) can be approximated in polynomial time with an approximation ratio of \(4  \frac 3k\).
M i n2G e n1E d g e can be approximated in polynomial time with an approximation ratio of 3.
5 Hardness Results
5.1 TSPInapproximability
In this section, we prove that Mind R e g1E d g e cannot be approximated better than MinT S P.
Theorem 5.1
For every d≥2, if M i ndR e g1E d g e can be approximated in polynomial time within a factor of r, then M i nT S P can be approximated in polynomial time within a factor of r.
Proof 26

\(\tilde w_{\{v_{i}, v_{j}\}} = 0\) for all v∈V, i≠j,

\(\tilde w_{\{u_{i}, v_{j}\}} = w_{\{u,v\}}\) for all u≠v, i and j.
Now assume that we have a connected dfactor R of H. We claim that we can construct a TSP tour T of G with \(w(T) \leq \tilde w(R)\). We construct a multiset \(T^{\prime }\) of edges of G as follows: For each edge {u _{ i },v _{ j }} of R, if u≠v, we add an edge {u,v} to \(T^{\prime }\). Otherwise, if u = v, we ignore the edge. The sum of the degrees in R of all vertices in each set V _{ v } is equal to (d+1)d and is therefore even. Thus, for each v, the number of edges leaving V _{ v } in R, which equals the number of edges incident to v in \(T^{\prime }\) by construction, is even as well. Since R is connected, the multigraph \(G^{\prime }=(V,T^{\prime })\) is connected as well. By construction, \(w(T^{\prime }) = \tilde w(R)\). Since \(G^{\prime }\) is connected and all its vertices have even degree, \(G^{\prime }\) is Eulerian. Therefore, we can obtain a TSP tour T from \(T^{\prime }\) by taking shortcuts. By the triangle inequality, \(w(T) \leq w(T^{\prime }) = \tilde w(R)\). □
The same construction as in the proof of Theorem 5.1 yields the same inapproximability result for M i nd R e g2E d g e.
M i nT S P is A P Xhard [25]. Furthermore, the reduction from M i nT S P to M i nd R e g1E d g e (and also to M i nd R e g2E d g e) is in fact an Lreduction [24] (see also Shmoys and Williamson [29, Section 16.2]). This proves the A P Xhardness of M i nd R e g1E d g e and M i nd R e g2E d g e for all d≥2.
5.2 Hardness for Growing d
In this section, we generalize the N Phardness proof by Cheah and Corneil [2] for the decision problem if a graph contains a connected dfactor to the case that d grows with n. We also extend Theorem 5.1 and the A P Xhardness to growing d.
 1.
We construct a gadget G _{ d+1} by removing a matching of size \(\lceil \frac {d}{2}\rceil 1\) from a complete graph on d+1 vertices.
 2.
We connect each vertex whose degree has been decreased by one to v.
The reduction itself takes a graph G for which we want to test if G contains a Hamiltonian cycle and maps it to a graph R _{ d }(G) as follows: For even d, R _{ d }(G) is the graph obtained by performing a dexpansion for every vertex of G. For odd d, the graph R _{ d }(G) is obtained by doing the following for each vertex v of G: add vertices u _{1},u _{2},…,u _{ d−2}; connect v to u _{1},…,u _{ d−2}; perform a dexpansion on u _{1},…,u _{ d−2}. We have that G contains a Hamiltonian cycle if and only if R _{ d }(G) contains a connected dfactor.
We note that R _{ d }(G) has (d+2)⋅n vertices for even d and Θ(d ^{2} n) vertices for odd d and can easily be constructed in polynomial time since d<n.
Theorem 5.2
For every fixed δ∈[0,1), there is a function d=d(n)=Θ(n ^{ δ }) that maps to even integers such that checking if an nvertex graph contains a connected d(n)factor is N Phard.
For every fixed \(\delta \in [0, \frac 12)\) , there is a function d=d(n)=Θ(n ^{ δ }) that maps to odd integers such that checking if an nvertex graph contains a d(n)factor is N Phard.
Proof 27
We first present the proof for the case that we map to even integers. After that, we briefly point out the difference for odd integers.
Let \(f = f(n) = 2 \lceil n^{\frac {\delta }{1\delta }} \rceil \) and apply R = R _{ f }(G). The graph R has g(n) = n⋅(f+2) vertices. since f is even. We have \(g = {\Theta }(n^{\frac 1{1\delta }})\). Now we determine d. We require d(g(n)) = f(n). This can be achieved because g = ω(n) is an injective function. From this, d=Θ(n ^{ δ }) follows. For natural numbers that are not images of g, we interpolate f to maintain the growth bound.
Let us now point out the differences for functions f mapping to odd integers. In this case, since the reduction for d maps to graphs of size Θ(d ^{2} n), we have to choose \(d = {\Theta }(n^{\frac {\delta }{1  2\delta }})\). This, however, works only up to δ<1/2. □
In the same way as the N Pcompleteness, the inapproximability can be transferred. The reduction creates graphs of size (d+1)⋅n. The construction is the same as in Section 5.1, and the proof follows the line of the proof of Theorem 5.2. Here, however, we do not have to distinguish between odd and even d for the symmetric variant, as the reduction in Section 5.1 is the same for both cases.
Theorem 5.3
For every fixed δ∈[0,1), there exists a function d=Θ(n ^{ δ }) such that M i ndR e g1E d g e and M i ndR e g2E d g e are A P Xhard and cannot be approximated better than Min  TSP.
Complementing Theorems 5.2 and 5.3, M i nd R e g1E d g e admits a PTAS for d≥n/3 and finding a connected dfactor can be done in polynomial time for d≥n/3 [23].
6 Conclusions and Open Problems
We conclude this paper with two questions for further research.
First, for edgeconnectivity, we require d≥2⌈k/2⌉. Since there exists an approximation algorithm for M i nk R e gk E d g e (for k≥2) [1], the only case for which it is unknown if a constant factor approximation algorithm exists is the generalized problem M i nk G e nk E d g e for odd values of k. We are particularly curious about approximation algorithms for M i n1G e n1E d g e, where we want to find a cheap connected graph with given vertex degrees. To get such algorithms, vertices with degree requirement 1 seem to be bothersome. (This seems to be a more general phenomenon in network design, as, for instance, the approximation algorithms by Fekete et al. [7] for boundeddegree spanning trees and by Fukunaga and Nagamochi [8] for kedgeconnected subgraphs with multiple edges both require that the minimum degree requirement is at least 2.) Still, we conjecture that constant factor approximation algorithms exist for these problems as well.
Second, we would like to see constant factor approximation algorithms for M i nd R e gk V e r t e x for the case k+1≤d≤2k−2 and for the general problem M i nd G e nk V e r t e x for k≤d≤2k−2. We conjecture that constant factor approximation algorithms exist for these problems.
Notes
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