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Further Results on Online Node- and Edge-Deletion Problems with Advice

  • Li-Hsuan Chen
  • Ling-Ju Hung
  • Henri Lotze
  • Peter Rossmanith
Conference paper
  • 521 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12126)

Abstract

In online edge- and node-deletion problems the input arrives node by node and an algorithm has to delete nodes or edges in order to keep the input graph in a given graph class at all times. We consider graph classes that can be characterized by forbidden sets of induced subgraphs and analyze the advice complexity of getting an optimal solution. We give almost tight lower and upper bounds for the Open image in new window , where there is one forbidden induced subgraph that may or may not be disconnected and tight bounds on the Open image in new window , where we have an arbitrary number of forbidden connected graphs. For the latter result we present an algorithm that computes the advice complexity directly from \(\mathcal{F}\). For the Open image in new window the advice complexity is basically an easy function of the size of the biggest component in H.

Keywords

Online algorithm Advice complexity Node deletion Edge deletion Delayed decision model Graph modification 

1 Introduction

Many classical online problems can be formulated as follows: Given an instance \(I = \{x_1,\ldots ,x_n\}\) as a series of elements ordered from \(x_1\) to \(x_n\), an algorithm receives them iteratively in this order, having to decide whether to include \(x_i\) into its solution at the point it receives it. It can base this decision only on the previously revealed \(x_1,\ldots ,x_{i-1}\) and must neither remove \(x_i\) from its solution later nor include any of the previously revealed elements into its solution. A way to measure the performance of such an online algorithm is the competitive ratio, which compares how much worse it performs compared to an optimal offline algorithm [4]. An algorithm is c-competitive if the competitive ratio of the algorithm is bounded by a constant c.

In most classical online problems such as the k-Server Problem, the Paging Problem or the Knapsack Problem as well as most other online problems, receiving the next \(x_i\) of an instance coincides with an algorithm having to process this request. This makes a lot of sense in the previously mentioned problems, but arguably less sense when there is no “need to act” after an item of the instance is presented, which may regularly happen in the instances of the problem that we study in this paper: Informally, the requests are single nodes of a graph that are iteratively revealed and our task is to keep the graph induced by these nodes free of a set \(\mathcal{F}\) of forbidden induced subgraphs by deleting nodes or edges. Obviously, there are sets and instances in which an arbitrary number of nodes can be revealed before any forbidden induced substructure is revealed. The offline variant of this problem was shown to be NP-Hard by Yannakakis [17].

In this work, we use a modified version, which we call the delayed decision model, which was already used in [16] and which is similar to the preemptive model used by Komm et al. [13]. We consider an instance \(I = \{x_1,\ldots ,x_n\}\) of an online minimization problem for which a solution \(S \subseteq I\) has to satisfy some condition C. Again, an algorithm ALG has to decide whether to include any element into its solution S. We denote the intermediate solution of an algorithm on an instance I at the revelation of element \(x_i\) – before the decision on whether to include it in S – by \(S_i^I(ALG)\). While in the classical definition, an algorithm has to decide on whether to include an element into its solution at the point of revelation, the algorithm may now wait until the condition C is violated by \(S_i^I(ALG)\). It may then include any of the previously revealed elements into its solution, but is still unable to revert any of its previous selections.

A selection of online problems that do not admit any algorithm with a constantly bounded competitive ratio, such as the Minimum Vertex Cover Problem and in particular general node and edge deletion problems are constantly competitive with delayed decision.

A simple example is the online Minimum Vertex Cover Problem. The input I is a series of induced subgraphs \(G[\{v_1\}], G[\{v_1,v_2\}], \ldots ,G[\{v_1,\ldots ,v_n\}]\) for which C states that \(S_i^I(ALG)\) is a vertex cover on \(G[\{v_1,\ldots ,v_i\}]\). In this setting, an algorithm has to include nodes into its current solution only once an edge is revealed that is not covered yet. While the Minimum Vertex Cover Problem is competitive in the maximum degree \(\varDelta \) of an input graph in the classical online setting [5], a competitive ratio of 2 can be proven for the delayed decision setting: The upper bound is given by always taking both nodes of an uncovered edge into the solution (this is the classical 2-approximation algorithm). The lower bound can be achieved by presenting an edge \(\{v_i,v_j\}\) and adding another edge to either \(v_i\) or \(v_j\), depending on which node is not taken into the solution by a deterministic online algorithm. If both nodes are taken into the solution then no additional edge is introduced. This gadget can be repeated and forces a deterministic algorithm to take two nodes into the vertex cover where one suffices.

We denote by H a finite graph and by \(\mathcal{F}\) a finite set of finite graphs. For a problem \(\varPi \) we denote the optimal solution size on an input I by \(opt_{\varPi }(I)\).

The competitive ratio is a standard method to analyze online algorithms and a relatively new alternative is the advice complexity introduced by Dobrev, Královič, and Pardubská [7], revised by Hromkovič, Královič and Královič [11] and refined by Böckenhauer et al. [2]. The advice complexity measures the amount of information about the future that is necessary to solve an online problem optimally or with a given competitive ratio. There is an oracle called “advisor” that knows the whole input instance and gives the online algorithm “advice” in the form of a binary string that can be read from a special advice tape. Many problems have been successfully analyzed in this model including the k-server problem [8], the Knapsack Problem [3], Job-Shop Scheduling [1] and many more. One criticism on the advice model is that in the real world such a powerful advisor usually cannot exist. However, the new research area of learning-augmented algorithms uses an AI-algorithm to guide classical algorithms to solve optimization problems and they are closely related to the advice complexity [14, 15]. A strong application of advice complexity are the lower bounds it provides: For example, the online knapsack problem can be solved with a competitive ratio of two by a randomized algorithm. It has been shown that this competitive ratio cannot be improved with \(o(\log n)\) advice bits.

We base our work on the definitions of advice complexity from [12] and [2], with a variation due to the modified online model we are working on: The length of the advice string is often measured as a function in the input length n, which usually almost coincides with the number of decisions an online algorithm has to make during its run. In the delayed decision model, the number of decisions may be smaller than n by a significant amount and we can measure the advice as \(f(opt_{\varPi }(I))\), i.e., a function of the size of the optimum solution. This usually does not work in classical online algorithms.

Tight results for the advice complexity of the Delayed Connected \(\mathcal{F}\)-Node Deletion Problem and of the Delayed Connected H-Edge Deletion Problem were shown in [16]. We show upper and lower bounds for the general Open image in new window and a tight bound for the Open image in new window . We leave open the exact advice complexity for the general Open image in new window and Delayed \(\mathcal{F}\)-Edge Deletion Problem, for which we can only provide lower bounds. Some proofs can be found only in the full version of this paper.

2 The \(\mathcal{F}\)-Node Deletion Problem and \(\mathcal{F}\)-Edge Deletion Problem Without Advice

For a graph \(G = (V,E)\) we write |G| to denote |V(G)| and ||G|| to denote |E(G)|. We use the symbol \(\trianglelefteq \) to denote an induced subgraph relation, i.e. \(A \trianglelefteq B\) iff A is an induced subgraph of B. We write \(\mathcal{G}\) to denote the set of all graphs.

We write \(G-U\) for \(G[V(G)-U]\) and \(G-u\) for \(G-\{u\}\) and also use \(G-E\) similarly for an edge set E. For graphs H and G we write \(H \trianglelefteq _\varphi G\) if there exists an isomorphism \(\varphi \) such that \(\varphi (H)\trianglelefteq G\). We call a set of graphs unordered if the members are pairwise maximal according to the induced subgraph relation \(\trianglelefteq \). It is easy to see that every Open image in new window can be reduced to one with an unordered \(\mathcal{F}\). A graph G is called \(\mathcal{F}\)-free if there is no \(H_i \trianglelefteq _\varphi G\) for any \(H_i \in \mathcal{F}\).

Definition 1

Let \(\mathcal {F}\) be an unordered set of graphs. Let I be a sequence of growing induced subgraphs \(G[\{v_1\}],\ldots , G[\{v_1,\ldots ,v_n\}]\). The Open image in new window is to delete a minimum size set of nodes S from G such that \(G-S\) is \(\mathcal{F}\)-free. We call \(S_i^I \subseteq \{v_1,\ldots ,v_i\}\) an (intermediate) solution for the Open image in new window on \(G[\{v_1,\ldots ,v_i\}]\) if \(G[\{v_1,\ldots ,v_i\}] - S_i^I\) is \(\mathcal{F}\)-free.

The Open image in new window is defined accordingly, with the condition C stating The Graph \(G[\{v_1,\ldots ,v_i\} - S_i^I(ALG)]\) is \(\mathcal{F}\)-free for all \(i \in \{1,\ldots ,n\}\) and some algorithm ALG. \(\mathcal{F}\)-Edge Deletion and Delayed \(\mathcal{F}\)-Edge Deletion are defined accordingly, with the solution being a set of edges. The graph is always revealed as a sequence of nodes. We will denote the Open image in new window for \(\mathcal{F}= \{H\}\) as the Open image in new window .

Lemma 1

There is at least one \(\mathcal{F}\) for which the Open image in new window is not c-competitive for any constant c.

Lemma 1 is not surprising. It generalizes that Vertex Cover admits no constantly bounded competitive ratio [5].

Lemma 2

There is at least one \(\mathcal{F}\) for which the \(\mathcal{F}\)-Edge Deletion Problem is not c-competitive for any constant c.

Lemma 3

The Open image in new window is k-competitive for \(k = \max _{H\in \mathcal{F}}\{|H|\}\). The Delayed \(\mathcal{F}\)-Edge Deletion Problem is k-competitive for \(k = \max _{H\in \mathcal{F}}\{||H||\}\).

Proof

Whenever an algorithm finds an induced H, it deletes all of its nodes, resp. edges.   \(\square \)

3 The Delayed H-Node Deletion Problem with Advice

If \(\mathcal{F}\) consists of connected subgraphs, tight results have already been proven in [16]. The advice complexity is exactly \({ opt}_\mathcal{F}(G)\log (|H|)+O(1)\) for a biggest graph \(H\in \mathcal{F}\). The problem becomes harder when the graphs in \(\mathcal{F}\) are disconnected and was left as an open question. We answer it partially by determining the advice complexity for the Open image in new window , where H can be disconnected.

Definition 2

Let \(C_G = \{C_1,C_2,\ldots ,C_j\}\) denote the set of components of G.

If a forbidden graph H is disconnected, it may contain multiple copies of the same component, e.g., three disjoint triangles among other components. If we were only to delete triangles, we would thus have to delete all but two copies to make the graph of an instance H-free. We introduce some notation to determine the number and the actual copies of a type of component.

Definition 3

Given a graph G. For a connected graph C we define the packing \(p_C(G)\) of C in G as the set of sets of pairwise node-disjoint copies of C in G and the packing number of C in G, \(\nu _{C}(G)\), as \(\max _{H \in p_C(G)}(|H|)\).

In other words, \(\nu _C(G)\) is the maximal number of C’s that can be packed node-disjointly into G.

We use the multiplicity of components in H in a lower bound that forces any algorithm to leave specific components such as the two specific triangles in our small example. To punish a wrong selection, we use a redundancy construction that maps a component C into a \(C'\) such that \(C \trianglelefteq C'\) and even \(C \trianglelefteq C'-\{v\}\) for every v holds, while \(C'\) does not contain two disjoint copies of C.

Definition 4

We call the graph \(H'\) a redundancy construction of a connected graph H with \(|H| > 1\) if there exists an isomorphism \(\varphi _1:\mathcal{G}\rightarrow \mathcal{G}\) such that for every isomorphism \(\varphi _2:\mathcal{G}\rightarrow \mathcal{G}\) the following holds:
  • \(\varphi _1(H) \trianglelefteq H' - v\) for all \(v \in V(H')\)

  • \(\varphi _1(H) \not \trianglelefteq H' - V(\varphi _2(H))\) if \(V(\varphi _2(H)) \subseteq V(H')\)

To show that such a redundancy construction actually exists, we use the following transformation.

Definition 5

Given a connected graph \(H = (V,E)\) with \(V = \{v_1,\ldots ,v_n\}\), \(n>1\), in some order and some \(k \in [2,n]\) s.t. \((v_1,v_k) \in E(H)\). \(H'\) is then constructed in the following way: \(V(H') = V(H) \cup \{v_i' \mid v_i \in V(H), i \ge 2\}\) and \(E(H') = E(H) \cup \{(v_i',v_j') \mid (v_i,v_j) \in E(H), v_i',v_j' \in V(H')\} \cup \{(v_1,v_i') \mid (v_1,v_i) \in E(H)\} \cup \{(v_k,v_j') \mid (v_1,v_j) \in E(H)\}\)

Intuitively, we create a copy of H except for a single node \(v_1\). The copied neighbors of \(v_1\) are then connected with \(v_1\). Lastly, some copied node is chosen and connected with the original neighbors of \(v_1\).

Example 1

A graph H and its redundancy construction \(H'\):

Lemma 4

The transformation in Definition 5 is a redundancy construction.

We denote an optimal solution of the Open image in new window on a graph G by \(sol_H(G)\).

3.1 Lower Bound

Theorem 1

Let H be a graph. Let \({C_{ max}}\) be a component of H of maximum size. Any online algorithm optimally solving the Open image in new window uses at least \({ opt}_H(G) \cdot \log |V({C_{ max}})| + (\nu _{{C_{ max}}}(H) - 1)\cdot \log ({ opt}_H(G))\) many advice bits on input G.

Proof

Let \(C_{H} = \{C_1,\ldots ,C_j\}\) and \(|V(C_1)| \le \ldots \le |V(C_j)|\). The adversary first presents \(k \ge \max \{\, \nu _{C_i}(H)\mid C_i\in C_H\,\}\) disjoint copies of each \(C_i \in C_{H}\) in an iterative way such that in each iteration one copy of each \(C_i\) is revealed node by node. If an algorithm deleted any nodes before an H is completed, the adversary would simply stop and the algorithm would not be optimal.

As soon as G is no longer H-free, any algorithm has to delete some node(s). For a \(C_i \in C_H\) it can either delete all \(C_i\) except for \(\nu _{C_i}(H) - 1\) occurrences and optionally some additional node(s). Obviously, deleting an additional node is not optimal, as the adversary would simply stop presenting nodes.

The following strategy will force an optimal online algorithm always to delete copies of \({C_{ max}}\). After all k copies of all \(C_i \in C_{H}\) are presented, additionally \(\max _{C_i \in C_{H}}\{\nu _{C_i}(H)\} - \nu _{{C_{ max}}}(H) + 1\) copies of each \(C_i \in C_{H} \setminus {C_{ max}}\) are presented. Deleting all \({C_{ max}}\) except for \(\nu _{{C_{ max}}}(H) - 1\) occurrences will thus only need \(k - \nu _{{C_{ max}}}(H) + 1\) deletions, while deleting any other component will need at least \(k - \max _{C_i \in C_{H}}\{\nu _{C_i}(H)\} + 1 + \max _{C_i \in C_{H}}\{\nu _{C_i}(H)\} - \nu _{{C_{ max}}}(H) + 1 = k - \nu _{{C_{ max}}}(H) + 2\) deletions. Thus, it is always optimal for any algorithm to focus on \({C_{ max}}\) for deletion.

After all components have been revealed - and some deletion(s) had to be made - a redundancy construction such as the one from Definition 5 is used in order to repair an arbitrary set of \(\nu _{{C_{ max}}}(H) - 1\) copies of \({C_{ max}}\). Every optimal algorithm will leave exactly \(\nu _{{C_{ max}}}(H) - 1\) copies of \({C_{ max}}\) after G is completely revealed. There are \({{{ opt}_H(G) + \nu _{{C_{ max}}}(H) - 1} \atopwithdelims (){{ opt}_H(G)}}\) many different ways to distribute the affected components onto all components and an algorithm without advice cannot distinguish them. In particular, each of these instances is part of a different, unique optimal solution, which deletes a node from all but the \(\nu _{{C_{ max}}}(H) - 1\) subgraphs. If an algorithm has chosen to delete a node from a component that is affected by the redundancy construction, this component is now repaired and demands an additional deletion. By definition, applying the redundancy construction does not result in additional disjoint copies of \({C_{ max}}\). Thus, it is still optimal to focus on \({C_{ max}}\) for deletion.

Finally, for every component that is not affected by a redundancy construction, the adversary glues a copy of \({C_{ max}}\) to one of its nodes as defined in [16]. It has \(|V({C_{ max}})|\) ways to do so for each copy of \({C_{ max}}\). Intuitively, the glueing operation joins two graphs by identifying a single node from both and connecting them by joining these two nodes into one.

We now measure how much advice an algorithm needs at least. First of all, it should be easy to see that the adversary is able to present \(|V({C_{ max}})|^{{ opt}_H(G)}\) many different instances regarding the deletion of nodes for the copies of \({C_{ max}}\) not selected for the redundancy construction.

Assuming \(\nu _{{C_{ max}}}(H) > 1\), any algorithm needs to determine the correct subset of \({ opt}_H(G)\) components out of \(k-1\) presented ones to delete one node from. As the adversary has \({{{ opt}_H(G) + \nu _{{C_{ max}}}(H) - 1} \atopwithdelims (){{ opt}_H(G)}}\) different ways to distribute these redundancies and since every single of these instances has a different unique optimal solution, any correct algorithm has to get advice on the complete distribution in the size of at least \(\log {{{ opt}_H(G) + \nu _{{C_{ max}}}(H) - 1} \atopwithdelims (){{ opt}_H(G)}} \ge (\nu _{{C_{ max}}}(H) - 1)\cdot \log ({ opt}_H(G))\) advice bits.   \(\square \)

3.2 Upper Bound

For simplicity of writing down the algorithm, we will assume in this section that we are only ever presented graphs which induce at least one forbidden subgraph H. Our algorithm can be easily transformed into one that only starts to read any advice once the first forbidden subgraph is completely revealed. For an instance with an online graph G with \(V(G) = \{v_1,\ldots ,v_n\}\) and a forbidden subgraph H, the advisor first computes the advice the algorithm is going to read during its run. It first identifies the set of components \(C_{ min\nu }\) which each require the fewest node deletions in G to make the graph H-free. Of these possible components, the advisor chooses the component with the fewest nodes which an optimal offline algorithm would choose, named \({C_{ min}}\) from here on. Finally, the advisor computes a list L of labels which will coincide with labels given by the algorithm to copies of \({C_{ min}}\) which are not to be deleted in an optimal solution. As there are at most \(\nu _{{C_{ min}}}(H)\cdot { opt}_H(G)\) node-disjoint copies of H in G and as Lemma 7 states that our algorithm uses at most \({ opt}_H(G) + O(1)\) labels, we can limit the range of possible labels by \([1,{ opt}_H(G)+O(1)]\). Finally, a number of advice bits is written for every deletion that the algorithm will make which encode the concrete node out of a copy of \({C_{ min}}\) is optimal to delete.

The algorithm starts by reading from the advice tape which component \({C_{ min}}\) to focus on for deletion and the list L, using self-delimiting encoding.

Whenever the next node \(x_i\) of the instance is revealed which fulfills \(\nu _{{C_{ min}}}(G_i) \ge \nu _{{C_{ min}}}(H)\), i.e. that there are at least as many node-disjoint copies of \({C_{ min}}\) in the current graph as in H, the algorithm will delete nodes from the graph as described in the following, otherwise the algorithm simply waits for the next node to be revealed.

To identify which node(s) of \(G_i\) are to be deleted, the algorithm first identifies all biggest sets of node-disjoint copies of \({C_{ min}}\). Of them it identifies a set P of which the most components have already received a label. Then all previously unlabeled copies of \({C_{ min}}\in P\) receive a new unique label. The algorithm now looks at the label list L given by the advisor. Every copy of \({C_{ min}}\in P\) whose label is not in L is now marked for deletion. The algorithm reads advice which concrete node out of every copy of \({C_{ min}}\) is optimal to delete.

Lemma 5

Algorithm 1 is correct.

Lemma 6

Algorithm 1 is optimal.

Definition 6

Given graphs GH and a labeling function \(l:\mathcal{G}\rightarrow \mathbf{N}\). We call a family \(\mathcal{C}\) of induced subgraphs of G a configuration, if every element of \(\mathcal{C}\) is isomorphic to H, \(l(C) \ne 0\) for each \(C\in \mathcal{C}\) and \(V(C_1) \cap V(C_2) = \emptyset \) for all \(C_1,C_2\in \mathcal{C}\), \(C_1\ne C_2\). The size of a configuration is the number of induced subgraphs it contains.

Informally speaking, a configuration is a set of disjoint induced subgraphs of G that already have a label.

Lemma 7

Given an online graph G, a forbidden graph H, as well as a subgraph \(C \in C_H\) of which there may be at most \(k = \nu _C(H) - 1\) disjoint copies present in G. Algorithm 1 assigns no more than \({ opt}_H(G) + O(1)\) labels to G if the advisor assigns \({C_{ min}}=C\) as specified in line 5.

Theorem 2

Let H be a graph. Let \(C_{min\nu } = \text {arg}\,\text {min}_{C \in C_H}\{\nu _{C}(G) - \nu _{C}(H)\}\) and \({C_{ min}}= \mathrm{{\text {arg}\,\text {min}}}_{C \in C_{min\nu }}\{|V(C)|\}\). The Open image in new window can be solved optimally using at most \({ opt}_H(G) \cdot \log |V({C_{ min}})| + O(\log { opt}_H(G))\) many advice bits on input G.

Proof

We count the number of advice bits used by Algorithm 1. We know by Lemma 5 and 6 that it is correct and optimal. The advice in line 5 is of constant size. As L only contains the labels for components which are not to be deleted and we limited the number of them by a constant in Lemma 7, only \(O(\log { opt}_H(G))\) advice, using self-delimiting encoding, is needed in line 6.

Finally, the algorithm reads advice on which node of each copy of \({C_{ min}}\) that is part of \(sol_H(G)\) to delete in line 21. This can be done using \({ opt}_H(G)\cdot \log |V({C_{ min}})|\) advice bits in total.   \(\square \)

4 The Delayed Connected \(\mathcal{F}\)-Edge Deletion Problem

Let \((d_1,\ldots ,d_k)\in \mathbf{N}^k\). Let m(n) be the solution to the recurrence relation
$$ m(n)= {\left\{ \begin{array}{ll} \sum \nolimits _{i=1}^k m(n-d_i) &{} \text {if } n\ge \max \{d_1,\ldots ,d_k\}\\ c_n &{} \text {otherwise} \end{array}\right. } $$
where \(c_n\ge 0\) and some \(c_i>0\) for \(0\le i< \max \{d_1,\ldots ,d_k\}\). Let \(\beta (d_1,\ldots ,d_k)=\inf _\tau \{\,\tau \mid m(n)=O(\tau ^n)\,\}\). Note that \(\beta \) does not depend on the \(c_i\)’s.

If \(S=\{D_1,\ldots ,D_k\}\) is a set of sets, then we define \(\beta (S)=\beta (|D_1|,\ldots ,|D_k|)\).

A homogeneous linear recurrence relation with constant coefficients usually has a solution of the form \(\varTheta (n^{k-1}\tau ^n)\) if \(\tau \) is the dominant singularity of the characteristic polynomial with multiplicity k [9]. However, here the coefficients of the characteristic polynomial are real numbers and there is exactly one sign change. By Descartes’ rule of signs there is exactly one positive real root and therefore its multiplicity has to be one [6, 10]. Therefore \(m(n)=\varTheta (\beta (S)^n)\).

Definition 7

Let \(\mathcal{F}\) be a set of forbidden connected induced subgraphs and \(H\in \mathcal{F}\). Let \(S\subseteq 2^{E(H)}\).

  1. 1.

    A set \(D\subseteq E(H)\) is H-optimal for a graph G if \(H\trianglelefteq G\) and \(G-D\) is \(\mathcal{F}\)-free and \({ opt}_\mathcal{F}(G)=|D|\).

     
  2. 2.

    A set \(D\subseteq E(H)\) is H-good for a graph G if \(H\trianglelefteq G\) and D is a non-empty subset of some \(\bar{D}\subseteq E(G)\) where \({ opt}_\mathcal{F}(G)=|\bar{D}|\) and \(G-\bar{D}\) is \(\mathcal{F}\)-free.

     
  3. 3.

    S is H-sound if \(H-D\) is \(\mathcal{F}\)-free for every \(D\in S\).

     
  4. 4.

    S is H-sufficient if for every connected graph G with \(H\trianglelefteq G\) there is a \(D\in S\) such that D is H-good for G.

     
  5. 5.

    S is H-minimal if for every \(D\in S\), there is a graph G such that D is H-good for G, but every \(D'\in S\), \(D'\ne D\) is not.

     

Lemma 8

Let \(\mathcal{F}=\{H_1,\ldots ,H_k\}\) be a set of connected graphs, G a graph and \(D\subseteq E(H_i)\) that is \(H_i\)-good for G. Then there is a subgraph \(G'\subseteq G\) such that D is \(H_i\)-optimal for \(G'\).

4.1 Upper Bound

Theorem 3

Let \(\mathcal{F}=\{H_1,\ldots ,H_k\}\) be a set of connected graphs and let \(S_i\) be \(H_i\)-sound and \(H_i\)-sufficient for all \(i \in \{1,\ldots ,k\}\). Then there is an \(m\in \mathbf{R}\) and an algorithm that solves the Open image in new window for every graph G with \(m\cdot { opt}_\mathcal{F}(G) +O(1)\) many advice bits where \(2^m\le \beta (S_i)\) for all \(i\in \{1,\ldots ,k\}\).

Proof

The algorithm receives \({ opt}_\mathcal{F}(G)\cdot \log (\max _i\{\beta (S_i)\})+O(1)\) many advice bits and then a graph G as a sequence of growing induced subgraphs. The algorithm interprets the advice as a number that can be between 0 and \(O((\max _i\{\beta (S_i)\})^{{ opt}_\mathcal{F}(G)})\).

The algorithm will delete in total exactly \({ opt}_\mathcal{F}(G)\) edges. We analyze the total number of different advice strings the algorithm might use when deleting \({ opt}_\mathcal{F}(G)\) edges.

When the algorithm receives a new node and its incident edges to form the next graph G it proceeds as follows: While G is not \(\mathcal{F}\)-free, choose some \(H_i\in \mathcal{F}\) for which \(H_i \trianglelefteq _\varphi G\). The advisor chooses one \(D\in S_i\) for which \(\varphi (D)\) is \(\varphi (H_i)\)-good for the graph at hand and puts it in the advice.

The advice strings are therefore partitioned into \(|S_i|\) subsets, one for each \(D\in S_i\). After deleting \(\varphi (D)\) the algorithm proceeds on the graph \(G-\varphi (D)\), where \({ opt}_\mathcal{F}(G)\) is now by |D| smaller. If \(m({ opt}_\mathcal{F}(G))\) is the total number of advice strings we get the recurrence \( m({ opt}_\mathcal{F}(G))=\max _i\bigl (\sum _{D\in S_i}m({ opt}_\mathcal{F}(G)-|D|)\bigr ) \) if \({ opt}_F(G)\) is at least as big as every \(D\in S_i\). Standard techniques show that \(m({ opt}_\mathcal{F}(G))=O(\max \{\beta (S_1),\ldots ,\beta (S_k)\}^{{ opt}_\mathcal{F}(G)})\).   \(\square \)

4.2 Lower Bound

Let \(\mathcal{F}=\{H_1,\ldots ,H_k\}\) be a set of connected graphs. We fix some correct algorithm A for the Open image in new window .

We define the sets \(S_i=S_i(A)\) for \(i=1,\ldots ,k\) as follows: \(D\in S_i\) if and only if there is some input sequence \(G_1,G_2,\ldots ,G_t\) such that algorithm A deletes the edge set \(D'\) from \(G_t\). Moreover, there is a set X and an isomorphism \(\varphi \) such that \(G[X]\cong H_i\), \(\varphi :V(H)\rightarrow X\), and \(\varphi (D)=D'\cap E(G[X])\). Informally speaking, the edge sets in \(S_i\) are those that are deleted from some isomorphic copy of \(H_i\) by algorithm A in some scenario.

We will need the following technical lemma. It states that we can find a matching with special properties in every connected bipartite graph. The matching should have the following properties. Let \(U'\) be the partners in the matching on top and \(V'\) on the bottom.

The first property is \(N(U')=V\), i.e., every node in V is connected to at least one node in \(U'\). The second property states that we have an induced matching, i.e., that the graph induced by \(U' \,\cup \, V'\) is a matching. The third property concerns the vertices in \(V'\): If \(v\in V'\) then N(v) contains several vertices from U, but exactly one node in \(U'\), i.e., its partner in the matching. We require that this partner is the smallest one in N(v).

Lemma 9

Let \(G=(U,V,E)\) be a bipartite graph where \(U=\{u_1,\ldots ,u_k\}\). Let \(\le \) be a preorder on U such that \(u_1\le \cdots \le u_k\). Moreover, assume that \(V\subseteq N(U)\), i.e., every node in V is connected to some node in U. Then there is a \(U'\subseteq U\) and \(V'\subseteq V\) such that
  1. 1.

    \(N(U')=V\),

     
  2. 2.

    \(G[U'\cup V']\) is a matching,

     
  3. 3.

    \(\min N(v)\in U'\) for every \(v\in V'\).

     

Lemma 10

Let \(\mathcal{F}=\{H_1,\ldots ,H_k\}\) be a set of connected graphs and \(S_i\) be \(H_i\)-sound and \(H_i\)-sufficient for all \(i \in \{1,\ldots ,k\}\). Then there are \(S'_i\subseteq S_i\) such that \(S'_i\) is \(H_i\)-sound, \(H_i\)-sufficient and \(H_i\)-minimal and moreover:

For every \(D'\in S'_i\) there is a graph G with \(H_i\trianglelefteq G\) such that \(D'\) is \(H_i\)-good for G and for every \(D\in S_i\setminus S'_i\) that is also \(H_i\)-good for G, it holds that \(|D|\ge |D'|\).

Theorem 4

Let \(\mathcal{F}=\{H_1,\ldots ,H_k\}\) be a set of connected graphs and assume that there is an algorithm A that can solve the Open image in new window for all inputs G with at most \(m\cdot { opt}_\mathcal{F}(G)+O(1)\) advice for some \(m\in \mathbf{R}\). Then there exist \(S'_i\) that are \(H_i\)-sound, \(H_i\)-sufficient, and \(H_i\)-minimal and \(\beta (S'_i)\le 2^m\) for every \(i\in \{1,\ldots ,k\}\).

Proof

By Lemma 10 there is an \(S'_i=\{D_1,\ldots ,D_r\}\subseteq S_i\) that is \(H_i\)-sound, \(H_i\)-sufficient, and \(H_i\)-minimal. It additionally has the property that for every \(D'\in S'_i\) there is a graph G with \(H_i\trianglelefteq G\) such that \(D'\) is \(H_i\)-good for G and for every \(D\in S_i\setminus S'_i\) that is also \(H_i\)-good for G, it holds that \(|D|\ge |D'|\).

Let \(l\in \mathbf{N}\). The adversary prepares \(\varTheta (\beta (S')^l)\) many instances by repeating the following procedure until the size of the optimum solution for the presented graph exceeds \(l-\max \{|D_1|,\ldots ,|D_r|\}\).

  1. 1.

    The adversary presents a disjoint copy of \(H_i\).

     
  2. 2.

    Then the adversary computes an induced supergraph \(G_j\) of \(H_i\) for which \(D_j\) is \(H_i\)-good, but all \(D_{j'}\in S'_i\) with \(j'\ne j\) are not \(H_i\)-good, for all \(1\le j\le r\). The existence of the graph \(G_j\) is guaranteed by the \(H_i\)-minimality of \(S'_i\). In particular there is a \(\bar{D}_j\supseteq D_j\) such that \(\bar{D}_j\) is \(H_i\)-optimal for \(G_j\). Let \(D_j'=\bar{D}_j-D_j\). Let \(G_j'=G_j-D_j'\). It is easy to see that \(D_j\) is \(H_i\)-optimal for \(G_j'\).

     

We show that no other \(D_{j'}\in S'_i\) is \(H_i\)-good for \(G_j'\). Assume otherwise. If \(D_{j'}\) is \(H_i\)-good for \(G_j'\) then there must be a \(\bar{D}_{j'}\supseteq D_{j'}\) that is \(H_i\)-optimal for \(G_j'\). Then \(G_j-D_{j'}-((\bar{D}_{j'}-D_{j'})\cup D_j')\) is \(\mathcal{F}\)-free. This implies that \(D_{j'}\) is \(H_i\)-good for \(G_j\) contradicting the \(H_i\)-minimality of \(S'_i\). Next the adversary transforms the \(H_i\) into one of the r possible \(G_j'\)s and presents the new vertices. Then \({ opt}_\mathcal{F}(G_j')=|D_j|\). Hence, the optimal solution size increases by \(|D_j|\).

In each round the input graph grows and the optimal solution size grows by \(|D_j|\). As soon as that size exceeds \(l-\max \{|D_1|,\ldots ,|D_r|\}\) the adversary keeps presenting disjoint copies of \(H_i\) without turning them into bigger connected graphs until the size reaches exactly l. The number N(l) of different instances is given by the following recurrence:
$$ N(l)= {\left\{ \begin{array}{ll} \sum \nolimits _{j=1}^r N(l-|D_j|)&{}\text {if }l\ge \max \{|D_1|,\ldots ,|D_r|\}\\ 1&{}\text {otherwise} \end{array}\right. } $$
It is easy to see that \(N(l)=\varTheta (\beta (S'_i)^l)\). The algorithm has to react differently on all of these instances: When the algorithm sees a new \(H_i\) to be turned into one of \(G'_1,\ldots ,G'_r\), it deletes different edge sets for each of the r possibilities.

The adversary constructed an instance that consists of a sequence of disjoint graphs \(G_{i_1}',\ldots ,G_{i_t}'\) from the set \(\{G_1',\ldots ,G_r'\}\) of which the total size is at least \(\sum \nolimits _{j=1}^t{ opt}_\mathcal{F}(G_{i_j})-\max \{|D_1|,\ldots ,|D_r|\}\) and O(1) many copies of \(H_i\). If G is the whole constructed instance we have \({ opt}_\mathcal{F}(G)=l+O(1)\) because \({ opt}_F(H_i)_\mathcal{F}=O(1)\). Together with \(N(l)=\varTheta (\beta (S'_i)^l)\) this means that Algorithm A uses at least \(\log N(l)=l\cdot \log \beta (S'_i)+O(1)={ opt}_\mathcal{F}(G)\log \beta (S_i')+O(1)\) advice bits. Assume Algorithm A uses at most \(m\cdot { opt}_\mathcal{F}(G)+O(1)\) advice bits on every graph G as stated in the precondition above. Then m cannot be smaller than \(\log \beta (S_i')\) for every \(i\in \{1,\ldots ,k\}\) because \({ opt}_\mathcal{F}(G)\) can be become arbitrarily big.   \(\square \)

Lemma 11

Let \(\mathcal{F}\) be a set of connected forbidden graphs, \(H\in \mathcal{F}\), and \(S\subseteq 2^{E(H)}\). There is an algorithm that can decide whether S is H-sufficient.

Proof

It is sufficient to verify for all connected graphs G with \(H\trianglelefteq G\) that some \(D\in S\) is H-good for G, i.e., there is an optimal solution for G that contains D. By Lemma 8 we can restrict our search to all such G’s that have an optimal solution that is a subset of E(H). There are infinitely many graphs G to check. To overcome this we define the unfolding of G, written \(\varUpsilon (G)\), as the set of the following graphs: Remember that \(H\trianglelefteq G\). If there is some \(H'\in \mathcal{F}\) with \(H'\trianglelefteq _\varphi G\) then \(G[E(H)\cup E(\varphi (H'))]\in \varUpsilon (G)\) (for every possible \(\varphi \)). If, however, \(\varUpsilon (G)\) contains two graphs \(G'\) and \(G''\) that are isomorphic via an isomorphism that is the identity on V(H), then only the lexicographically smaller one is retained.

This means that the unfolding of G contains all induced subgraphs that consist of H and one other copy of some forbidden induced subgraph from \(\mathcal{F}\) that must overlap with H in some way (because we assumed that G has an optimal solution that consists solely of edges from H). Here is a small example: Let Open image in new window , Open image in new window , Open image in new window . Then Open image in new window .

It is easy to see that deleting some \(D\subseteq E(H)\) from G makes it \(\mathcal{F}\)-free iff deleting the same D from all graphs \(G'\in \varUpsilon (G)\) makes all these \(G'\) \(\mathcal{F}\)-free. Hence, there is an optimal solution for G that is a subset of E(H) iff there is such a subset that is “optimal” for \(\varUpsilon (G)\) (i.e., deletion of no smaller edge set can make all graphs in \(\varUpsilon (G)\) \(\mathcal{F}\)-free).

There are only finitely possibilities for \(\varUpsilon (G)\) and we can enumerate all of them. Let us say this enumeration is \(\varUpsilon _1,\ldots ,\varUpsilon _t\). For each \(\varUpsilon _i\) we first find out, whether there is a G with \(\varUpsilon (G)=\varUpsilon _i\). We can do this by enumerating all graphs G up to a size that does not exceed the sum of the sizes of all graphs in \(\varUpsilon _i\) and computing \(\varUpsilon (G)\) for them. If indeed \(\varUpsilon (G)=\varUpsilon _i\) then we test whether S is H-good for G. Iff these tests pass for all i then S is indeed H-sufficient.   \(\square \)

Theorem 5

Let \(\mathcal{F}=\{H_1,\ldots ,H_k\}\) be connected graphs. The advice complexity for Delayed Connected \(\mathcal F\)-Edge Deletion is \(m\cdot { opt}_\mathcal{F}(G)+O(1)\) where \( m=\max _{i\in \{1,\ldots ,k\}} \min \{\,\log \beta (S)\mid S\subseteq 2^{E(H)}, S \text { is } H_i\text {-sound and }H_i\text {-sufficient}\,\}. \) There is an algorithm that can compute m from \(\mathcal{F}\). More specifically, there is an algorithm that gets \(\mathcal{F}\) and \(t\in \mathbf{N}\) as the input and returns the tth bit of the binary representation of m.

Proof

\(\le \)” by Theorem 3. “\(\ge \)” by Theorem 4. An algorithm can enumerate all possible \(S\subseteq E(H)\) and then test if S is \(H_i\)-sound and \(H_i\)-sufficient (by Lemma 11). Then \(\beta (S)\) is computed by finding the only real root of the characteristic polynomial of the corresponding recurrence relations [9].   \(\square \)

Notes

Acknowledgement

We like to thank Ratislav Královič for helping significantly to simplify the proof of Lemma 7.

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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Li-Hsuan Chen
    • 1
  • Ling-Ju Hung
    • 2
  • Henri Lotze
    • 3
  • Peter Rossmanith
    • 3
  1. 1.Kenkone Medical Co.TaipeiTaiwan
  2. 2.National Taipei University of BusinessTaipeiTaiwan
  3. 3.RWTH Aachen UniversityAachenGermany

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