Incremental Optimization of Independent Sets under Reachability Constraints

04/25/2018 ∙ by Takehiro Ito, et al. ∙ University of Waterloo Tohoku University 0

We introduce a new framework for reconfiguration problems, and apply it to independent sets as the first example. Suppose that we are given an independent set I_0 of a graph G, and an integer l > 0 which represents a lower bound on the size of any independent set of G. Then, we are asked to find an independent set of G having the maximum size among independent sets that are reachable from I_0 by either adding or removing a single vertex at a time such that all intermediate independent sets are of size at least l. We show that this problem is PSPACE-hard even for bounded pathwidth graphs, and remains NP-hard for planar graphs. On the other hand, we give a linear-time algorithm to solve the problem for chordal graphs. We also study the fixed-parameter (in)tractability of the problem with respect to the following three parameters: the degeneracy d of an input graph, a lower bound l on the size of the independent sets, and a lower bound s on the solution size. We show that the problem is fixed-parameter intractable when only one of d, l, and s is taken as a parameter. On the other hand, we give a fixed-parameter algorithm when parameterized by s+d; this result implies that the problem parameterized only by s is fixed-parameter tractable for planar graphs, and for bounded treewidth graphs.

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1 Introduction

Recently, the reconfiguration framework [10] has been intensively applied to a variety of search problems. (See, e.g., surveys [8, 19].) For example, the independent set reconfiguration problem is one of the most well-studied reconfiguration problems [1, 2, 3, 9, 11, 12, 13, 15, 16, 17, 22, 23]. For a graph , a vertex subset is an independent set of if no two vertices in are adjacent in . Suppose that we are given two independent sets and of , and imagine that a token (coin) is placed on each vertex in . Then, for an integer lower bound , independent set reconfiguration under the rule is the problem of determining whether we can transform into via independent sets of size at least such that each intermediate independent set can be obtained from the previous one by either adding or removing a single token.111 stands for Token Addition and Removal, and there are two other well-studied reconfiguration rules called (Token Sliding) and (Token Jumping) [13]. We omit the details in this paper. In the example of Figure 1, can be transformed into via the sequence , but not into , when .

Figure 1: A sequence of independent sets under the rule for the lower bound , where the vertices in independent sets are colored with black.

Like this problem, many reconfiguration problems have the following basic structure: we are given two feasible solutions of an original search problem, and are asked to determine whether we can transform one into the other by repeatedly applying a specified reconfiguration rule while maintaining feasibility. These kinds of reconfiguration problems model several “dynamic” situations of systems, where we wish to find a step-by-step transformation from the current configuration of a system into a more desirable one.

However, it is not easy to obtain a more desirable configuration for an input of a reconfiguration problem, because many original search problems are NP-hard. Furthermore, there may exist (possibly, exponentially many) desirable configurations; even if we can not reach a given target from the current configuration, there may exist another desirable configuration which is reachable. Recall the example of Figure 1, where both and have the same size three (which is larger than that of the current independent set ), but can reach only .

1.1 Our problem

In this paper, we propose a new framework for reconfiguration problems which asks for a more desirable configuration that is reachable from the current one. As the first application of this new framework, we consider independent set reconfiguration because it is one of the most well-studied reconfiguration problems.

Suppose that we are given a graph , an integer lower bound , and an independent set of . Then, we are asked to find an independent set of such that is maximized and can be transformed into under the rule for the lower bound . We call this problem the optimization variant of independent set reconfiguration (denoted by Opt-ISR). To avoid confusion, we call the standard independent set reconfiguration problem the reachability variant (denoted by Reach-ISR).

Note that is not always a maximum independent set of the graph . For example, the graph in Figure 1 has a unique maximum independent set of size four (consisting of the vertices on the left side), but cannot be transformed into it. Indeed, for this example when .

1.2 Related results

Although Opt-ISR is being introduced in this paper, some previous results for Reach-ISR are related in the sense that they can be converted into results for Opt-ISR. We present such results here, explaining their relation to our results in Sections 24, after formally defining Opt-ISR and notation.

Ito et al. [10] showed that Reach-ISR under the rule is PSPACE-complete. On the other hand, Kamiński et al. [13] proved that any two independent sets of size at least are reachable under the rule with the lower bound for even-hole-free graphs.

Reach-ISR has been studied well from the viewpoint of fixed-parameter (in)tractability. Mouawad et al. [17] showed that Reach-ISR under the rule is W[1]-hard when parameterized by the lower bound and the length of a desired sequence (i.e., the number of token additions and removals). Lokshtanov et al. [16] gave a fixed-parameter algorithm to solve Reach-ISR under the rule when parameterized by the lower bound and the degeneracy of an input graph.

1.3 Our contributions

Figure 2: Our results with respect to graph classes.
(no parameter) lower bound solution size
(no parameter) NP-hard for fixed W[1]-hard for , XP
(i.e., no FPT, no XP) (i.e., no FPT)
[Corollary 3.1] [Theorems 4.1, 4.1]
degeneracy PSPACE-hard for fixed NP-hard for fixed FPT
(i.e., no FPT, no XP) (i.e., no FPT, no XP)
[Theorem 3.2] [Corollary 3.1] [Theorem 4.2]
Table 1: Our results with respect to parameters.

In this paper, we study the problem from the viewpoints of polynomial-time solvability and fixed-parameter (in)tractability.

We first study the polynomial-time solvability of Opt-ISR with respect to graph classes, as summarized in Figure 2. More specifically, we show that Opt-ISR is PSPACE-hard even for bounded pathwidth graphs, and remains NP-hard even for planar graphs. On the other hand, we give a linear-time algorithm to solve the problem for chordal graphs. We note that our algorithm indeed works in polynomial time for even-hole-free graphs (which form a larger graph class than that of chordal graphs) if the problem of finding a maximum independent set is solvable in polynomial time for even-hole-free graphs; currently, its complexity status is unknown.

We next study the fixed-parameter (in)tractability of Opt-ISR, as summarized in Table 1. In this paper, we consider mainly the following three parameters: the degeneracy of an input graph, a lower bound on the size of the independent sets, and the solution size . As shown in Table 1, we completely analyze the fixed-parameter (in)tractability of the problem according to these three parameters; details are explained below.

We first consider the problem parameterized by a single parameter. We show that the problem is fixed-parameter intractable when only one of , , and is taken as a parameter. In particular, we prove that Opt-ISR is PSPACE-hard for a fixed constant and remains NP-hard for a fixed constant , and hence the problem does not admit even an XP algorithm for each single parameter or under the assumption that or . On the other hand, Opt-ISR is W[1]-hard for , and admits an XP algorithm with respect to .

We thus consider the problem taking two parameters. However, the problem still remains NP-hard for a fixed constant , and hence it does not admit even an XP algorithm for under the assumption that . Note that the combination of and is meaningless, since , as explained in Section 4. On the other hand, we give a fixed-parameter algorithm when parameterized by ; this result implies that Opt-ISR parameterized only by is fixed-parameter tractable for planar graphs, and for bounded treewidth graphs.

2 Preliminaries

In this paper, we consider only simple graphs, without loss of generality. For a graph , we denote by and the vertex set and edge set of , respectively. For a vertex , let , and let . The set is called the (open) neighborhood of in , while is called the closed neighborhood of in . For a graph and a vertex subset , denotes the subgraph of induced by , that is, and . For a vertex subset , we simply write to denote . We denote by the symmetric difference between two sets and , that is, .

Optimization variant of independent set reconfiguration

We now formally define our problem. For an integer and two independent sets and of a graph such that and , a sequence of independent sets of is called a reconfiguration sequence between and under the rule if satisfies the following three conditions (a)–(c):

  • and ;

  • is an independent set of size at least for each ; and

  • for each .

To emphasize the lower bound on the size of any independent set, we sometimes write instead of . Note that any reconfiguration sequence is reversible, that is, is a reconfiguration sequence between and under the rule. We say that two independent sets and are reachable under the rule if there exists a reconfiguration sequence between and under the rule. We write if and are reachable under the rule.

Our problem aims to optimize a given independent set under the rule. Specifically, the optimization variant of independent set reconfiguration (Opt-ISR for short) is defined as follows:

  • A graph , an integer , and an independent set of such that

  • Find an independent set of such that and is maximized.

We denote by a triple an instance of Opt-ISR, and call a desired independent set of a solution to . Note that a given independent set may itself be a solution. Opt-ISR simply outputs a solution to , and does not require the specification of an actual reconfiguration sequence from to the solution.

We close this section with noting the following observation which says that Opt-ISR for an instance is equivalent to finding a maximum independent set of . Every maximum independent set of a graph is a solution to an instance of Opt-ISR, where is any independent set of .

Proof.

Because the lower bound on the size of the independent sets is set to zero, any (maximum) independent set of is reachable from any independent set of , as follows: We first remove all vertices in one by one, and then add all vertices in one by one. Thus, the lemma follows. ∎

3 Polynomial-Time Solvability

In this section, we study the polynomial-time solvability of Opt-ISR.

3.1 NP-hardness for planar graphs

Lemma 2 implies that results for the maximum independent set problem can be applied to Opt-ISR for . For example, we have the following theorem, because maximum independent set remains NP-hard even for planar graphs [7]. Opt-ISR is NP-hard for planar graphs and , where is a lower bound on the size of the independent sets.

For an integer , a graph is -degenerate if every induced subgraph of has a vertex of degree at most  [14]. The degeneracy of is the minimum integer such that is -degenerate. It is known that the degeneracy of any planar graph is at most five [14], and hence we have the following corollary. Opt-ISR is NP-hard for -degenerate graphs and , where is a lower bound on the size of the independent sets.

This corollary implies that Opt-ISR admits neither a fixed-parameter algorithm nor an XP algorithm when parameterized by under the assumption that , where is an upper bound on the degeneracy of an input graph and is a lower bound on the size of the independent sets. We will discuss the fixed parameter (in)tractability of Opt-ISR more deeply in Section 4.

3.2 PSPACE-hardness for bounded pathwidth graphs

In this subsection, we show that Opt-ISR is PSPACE-hard even if the pathwidth of an input graph is bounded by a constant. We first define the pathwidth of a graph, as follows [20]. A path-decomposition of a graph is a sequence of vertex subsets of such that

  • for each vertex of , there exists a subset such that ;

  • for each edge of , there exists a subset such that ; and

  • for any three indices such that , holds.

The pathwidth of is the minimum value such that there exists a path-decomposition of for which holds for all . A bounded pathwidth graph is a graph whose pathwidth is bounded by a constant.

The following theorem is the main result of this subsection. Opt-ISR is PSPACE-hard for bounded pathwidth graphs.

Proof.

We give a polynomial-time reduction from the (reachability variant of) maximum independent set reconfiguration problem (MISR for short), defined as follows [22]:

  • A graph , and two maximum independent sets and of

  • Determine whether or not, where

We denote by a triple an instance of MISR. This problem is known to be PSPACE-complete for bounded bandwidth graphs [22]. Since the pathwidth of a graph is at most the bandwidth of the graph [21], MISR is PSPACE-complete also for bounded pathwidth graphs.

Figure 3: (a) Graph and its independent set for MISR, and (b) the corresponding graph for Opt-ISR, where newly added edges are depicted by thick dotted lines.

Let be an instance of MISR such that the pathwidth of is bounded by a constant. Then, we construct a corresponding instance of Opt-ISR, as follows. (See also Figure 3.) We add a new vertex to the graph , and join it with all vertices in ; let be the resulting graph, that is, and . Since the pathwidth of is bounded by a constant and , the pathwidth of is also bounded by a constant. Let , and . This completes the construction of the corresponding instance of Opt-ISR. This construction can be done in polynomial time.

We now prove the correctness of our reduction. We first claim that has only one maximum independent set, and it is of size . To see this, recall that is a maximum independent set of . Therefore, if has an independent set such that , it must contain . Since is adjacent to all vertices in , only can be such an independent set of size , as claimed. Therefore, to complete the correctness proof of our reduction, we prove that Opt-ISR for outputs if and only if on . Since is only the maximum independent set in , we indeed prove that on if and only if on .

We first prove the if direction. Suppose that on . Since contains as an induced subgraph, we have on . Then, can be obtained simply by adding to , and hence we can conclude that on .

We next prove the only-if direction. Suppose that on , that is, there exists a reconfiguration sequence on under the rule. Let be the first independent set in which contains ; notice that because we know . Since is an independent set of , no vertex in is adjacent to . By the construction of , we thus have . Since appears in , we know . Therefore, we can construct a reconfiguration sequence between and under the rule by simply adding the vertices in one by one. Then, by combining and serially, we can obtain a reconfiguration sequence between and under the rule such that all independent sets in the sequence do not contain . We thus have on . ∎

3.3 Linear-time algorithm for chordal graphs

A graph is chordal if every induced cycle in is of length three [4]. The main result of this subsection is the following theorem. Opt-ISR is solvable in linear time for chordal graphs.

This theorem can be obtained from the following lemma; we note that a maximum independent set of a chordal graph can be found in linear time [6], and the maximality of a given independent set can be checked in linear time. Let be an instance of Opt-ISR such that is a chordal graph, and let be any maximum independent set of . Then, a solution to can be obtained as follows:

Proof.

We first consider the case where is a maximal independent set of and . In this case, we cannot remove any vertex from because . Furthermore, since is maximal, we cannot add any vertex in to while maintaining independence. Therefore, has no independent set which is reachable from , and hence .

We then consider the other case, that is, is not a maximal independent set of or . Observe that it suffices to consider the case where holds; if , then is not maximal and hence we can obtain an independent set of such that and by adding some vertex in . To prove , we below show that holds if .

Let be any independent set of size . Then, holds, because we can obtain from by removing vertices in one by one. Similarly, let be any independent set of size ; we know that holds. Kamiński et al. [13] proved that any two independent sets of the same size are reachable under the rule for even-hole-free graphs. Since any chordal graph is even-hole free, we thus have . Therefore, we have , and hence we can conclude that holds as claimed. ∎

We note that Lemma 3.3 indeed holds for even-hole-free graphs, which contain all chordal graphs. However, the complexity status of the (ordinary) maximum independent set problem is unknown for even-hole-free graphs, and hence we do not know if we can obtain in polynomial time. Indeed, Theorem 2 implies that the complexity status of Opt-ISR also remains open for even-hole-free graphs.

4 Fixed Parameter Tractability

In this section, we study the fixed parameter (in)tractability of Opt-ISR. We take the solution size of Opt-ISR as the parameter. More formally, for an instance , the problem Opt-ISR parameterized by solution size asks whether has an independent set such that and . We may assume that ; otherwise it is a -instance because itself is a solution. We sometimes denote by a -tuple an instance of Opt-ISR parameterized by solution size .

4.1 Single parameter: solution size

We first give an observation that can be obtained from independent set. Because independent set is W[1]-hard when parameterized by solution size  [18], Lemma 2 implies the following theorem. Opt-ISR is -hard when parameterized by solution size .

This theorem implies that Opt-ISR admits no fixed-parameter algorithm with respect to solution size under the assumption that . However, it admits an XP algorithm with respect to , as in the following theorem. For an instance , Opt-ISR parameterized by solution size can be solved in time , where is the number of vertices in .

Proof.

We construct an auxiliary graph , defined as follows: Each node in corresponds to an independent set of such that , and there is an edge in between two nodes corresponding to independent sets and if and only if holds. Notice that has a node corresponding to , since . Then, by breadth-first search starting from the node corresponding to , we can check if there is an independent set of such that and .

We now estimate the running time of the algorithm. Let

and denote the numbers of vertices and edges in , respectively. The number of (candidates of) nodes in can be bounded by . For each enumerated vertex subset of , we check if it forms an independent set of ; this can be done in time . Therefore, the node set can be constructed in time . We then check each pair of nodes in ; there are pairs. We join the pair by an edge in if their corresponding independent sets differ in only one vertex; we can check this condition in time for each pair of nodes. In this way, we can construct the auxiliary graph in time in total. Since breadth-first search can be executed in time , our algorithm runs in time in total. ∎

4.2 Two parameters: solution size and degeneracy

As we have shown in Theorem 4.1, Opt-ISR admits no fixed-parameter algorithm when parameterized by the single parameter of solution size under the assumption that . In addition, Theorem 3.2 implies that the problem remains PSPACE-hard even if the degeneracy of an input graph is bounded by a constant, and hence Opt-ISR does not admit even an XP algorithm with respect to the single parameter under the assumption that . In this subsection, we take these two parameters, and develop a fixed-parameter algorithm as in the following theorem. Opt-ISR admits a fixed-parameter algorithm when parameterized by , where is the solution size and is the degeneracy of an input graph.

Before proving the theorem, we note the following corollary which holds for planar graphs, and for bounded pathwidth graphs. Recall that Opt-ISR is intractable (from the viewpoint of polynomial-time solvability) for these graphs, as shown in Theorems 3.1 and 3.2. Opt-ISR parameterized by solution size is fixed-parameter tractable for planar graphs, and for bounded treewidth graphs.

Proof.

Recall that the degeneracy of any planar graph is at most five. It is known that the degeneracy of a graph is at most the treewidth of the graph. Thus, the corollary follows from Theorem 4.2. ∎

4.2.1 Outline of algorithm

As a proof of Theorem 4.2, we give such an algorithm. We first explain our idea and the outline of the algorithm. Our idea is to extend a fixed-parameter algorithm for Reach-ISR when parameterized by  [16].

If an input graph consists of only a fixed-parameter number of vertices, then we apply Theorem 4.1 to the instance and obtain the answer in fixed-parameter time (Lemma 4.2.2). We here use the fact (stated by Lokshtanov et al. [16, Proposition 3]) that a -degenerate graph consists of a small number of vertices if it has a small number of low-degree vertices (Lemma 4.2.2).

Therefore, it suffices to consider the case where an input graph has many low-degree vertices. In this case, we will kernelize the instance: we will show that there always exists a low-degree vertex which can be removed from an input graph without changing the answer ( or ) to the instance. Our kernelization has two stages. In the first stage, we focus on “twins” (two vertices that have the same closed neighborhoods), and prove that one of them can be removed without changing the answer (Lemma 4.2.3). The second stage will be executed only when the first stage cannot kernelize the instance to a sufficiently small size. The second stage is a bit involved, and makes use of the Sunflower Lemma by Erdös and Rado [5].

4.2.2 Graphs having a small number of low-degree vertices

We now give our algorithm. Suppose that is an instance of Opt-ISR parameterized by solution size such that is a -degenerate graph. We assume that ; otherwise is a -instance because itself is a solution.

We first show the following property for -degenerate graphs, which is a little bit stronger claim than that of Lokshtanov et al. [16, Proposition 3]; however, the proof is almost the same as that of [16]. Suppose that a graph is -degenerate, and let be the set of all vertices of degree at most in . Then, .

Proof.

Suppose for a contradiction that holds for some integer . Then, , and hence we have

This contradicts the fact that holds for any -degenerate graph  [14]. ∎

Let , and let . We introduce a function which depends on only and ; more specifically, let . We now consider the case where has only a fixed-parameter number of vertices of degree at most . If , then Opt-ISR can be solved in fixed-parameter time with respect to and .

Proof.

Since and , we have . By Lemma 4.2.2 we thus have . Therefore, depends on only and . Then, this lemma follows from Theorem 4.1. ∎

4.2.3 First stage of kernelization

We now consider the remaining case, that is, holds. The first stage of our kernelization focuses on “twins”, two vertices having the same closed neighborhoods, and removes one of them without changing the answer. Suppose that there exist two vertices and in such that . Then, is a -instance if and only if is.

Proof.

We note that and , because . Then, the if direction is trivial, and hence we prove the only-if direction. Suppose that is a -instance, and hence has an independent set such that and . Then, there exists a reconfiguration sequence . Since , we know that and are adjacent in and hence no independent set of contains and at the same time. We now consider a new sequence defined as follows: for each , let

Since each , , is an independent set of and , each forms an independent set of . In addition, since for all , we have . Therefore, is a reconfiguration sequence such that no independent set in contains . Since , we can conclude that is a -instance. ∎

We repeatedly apply Lemma 4.2.3 to a given graph, and redefine as the resulting graph; we also redefine and according to the resulting graph . Then, any two vertices and in satisfy . If , then we have completed our kernelization; recall Lemma 4.2.2. Otherwise, we will execute the second stage of our kernelization described below.

4.2.4 Second stage of kernelization

In the second stage of the kernelization, we use the classical result of Erdös and Rado [5], known as the Sunflower Lemma. We first define some terms used in the lemma. Let be non-empty sets over a universe , and let which may be an empty set. Then, the family is called a sunflower with a core if holds for each , and holds for each satisfying . The set is called a petal of the sunflower. Note that a family of pairwise disjoint sets always forms a sunflower (with an empty core). Then, the following lemma holds. [Erdös and Rado [5]] Let be a family of sets (without duplicates) over a universe such that each set in is of size at most . If , then there exists a family which forms a sunflower having petals. Furthermore, can be computed in time polynomial in , , and .

We now explain the second stage of our kernelization, and make use of Lemma 4.2.4. Let denote the vertices in , and let be the set of closed neighborhoods of all vertices in . In the second stage, recall that holds for any two vertices and in , and hence no two sets in are identical. We set . Since each is of degree at most in , each is of size at most . Notice that . Therefore, we can apply Lemma 4.2.4 to the family by setting and , and obtain a sunflower with a core and petals in time polynomial in , , and . Let be the set of vertices whose closed neighborhoods correspond to the sunflower , that is, . The following lemma says that at least vertices in are contained in the petals of the sunflower (i.e., not in the core ), and they forms an independent set of . is an independent set of , and forms a clique in . Furthermore, .

Proof.

To prove the lemma, we first claim that two distinct vertices are adjacent in if and only if . Recall that their closed neighborhoods and belong to the sunflower with the core , and hence holds. In the if direction proof of the claim, since , we have and hence and are adjacent in . On the other hand, in the only-if direction proof of the claim, since and are adjacent in , we have . In this way, the claim holds.

This claim indeed implies that is an independent set of , and forms a clique in . Therefore, to complete the proof of this lemma, it suffices to prove that holds. To see this, notice that holds, because otherwise forms a clique in of size at least ; this contradicts the assumption that is a -degenerate graph. Then, . ∎

We are now ready to give the following lemma, as the second stage of the kernelization. Let be any vertex in . Then, is a -instance if and only if is.

Proof.

Note that since . Then, the if direction clearly holds, and hence we prove the only-if direction. Suppose that is a -instance, and hence has an independent set such that and . Then, there exists a reconfiguration sequence . If no independent set in contains , then the only-if direction holds. Therefore, we consider the case where at least one independent set in contains . Let be the first independent set in which contains , that is, for all . We assume without loss of generality that holds for all .

Figure 4: Illustration for Lemma 4.2.4, where two vertices are depicted by squares. By the definition of a sunflower, all vertices adjacent to both and must be contained in .

Let , where . We now claim that is an independent set of such that and on ; then is a -instance. Since , Lemma 4.2.4 says that is an independent set of . Furthermore, since does not contain any vertex in , is an independent set of . Recall that holds on , and hence we know . Then, holds on by adding the vertices in to one by one. We finally prove that by showing that holds. Since (by Lemma 4.2.4) and , it suffices to prove that holds for each vertex . Since is obtained by adding to , we know . Since , we thus have . Therefore, each vertex is adjacent to at most one vertex in , because otherwise must be contained in . (See also Figure 4.) ∎

We can repeatedly apply Lemma 4.2.4 to until the resulting graph has at most vertices of degree at most . Then, by Lemma 4.2.2 we have completed our kernelization.

This completes the proof of Theorem 4.2.

5 Conclusions

In this paper, we have introduced a new framework for reconfiguration problems, and applied it to independent set reconfiguration. As shown in Figure 2 and Table 1, we have studied the problem from the viewpoints of polynomial-time solvability and the fixed-parameter (in)tractability, and shown several interesting contrasts among graph classes and parameters. In particular, we gave a complete analysis of the fixed-parameter (in)tractability with respect to the three parameters.

Acknowledgments

We thank Tatsuhiko Hatanaka for his insightful suggestions on this new framework. We are grateful to Tesshu Hanaka, Benjamin Moore, Vijay Subramanya, and Krishna Vaidyanathan for valuable discussions with them. Research by Japanese authors is partially supported by JST CREST Grant Number JPMJCR1402, and JSPS KAKENHI Grant Numbers JP16K00004 and JP17K12636, Japan. Research by Naomi Nishimura is supported by the Natural Science and Engineering Research Council of Canada.

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