The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies

Gopalan, Parikshit; Kolaitis, Phokion G.; Maneva, Elitza; Papadimitriou, Christos H.

doi:10.1137/07070440x

Cited by 135 publications

(192 citation statements)

References 28 publications

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“…It is very interesting to compare the work presented in this paper and [2,3] with [8], which contains remarkably similar results. For a given instance ϕ of the Boolean satisfiability problem, the authors of [8] define the graph G(ϕ) as the graph with vertex set the satisfying assignments of ϕ, and assignments adjacent whenever they differ in exactly one bit.…”

Section: Theorem 2 ([4]) Let G Be a 3-colourable Graph With N Verticsupporting

confidence: 67%

Finding Paths between graph colourings: PSPACE-completeness and superpolynomial distances

Bonsma

Cereceda

2009

Theoretical Computer Science

131

134

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a b s t r a c tSuppose we are given a graph G together with two proper vertex k-colourings of G, α and β. How easily can we decide whether it is possible to transform α into β by recolouring vertices of G one at a time, making sure we always have a proper k-colouring of G? This decision problem is trivial for k = 2, and decidable in polynomial time for k = 3. Here we prove it is PSPACE-complete for all k ≥ 4. In particular, we prove that the problem remains PSPACE-complete for bipartite graphs, as well as for: (i) planar graphs and 4 ≤ k ≤ 6, and (ii) bipartite planar graphs and k = 4. Moreover, the values of k in (i) and (ii) are tight, in the sense that for larger values of k, it is always possible to recolour α to β.We also exhibit, for every k ≥ 4, a class of graphs {G N,k : N ∈ N * }, together with two k-colourings for each G N,k , such that the minimum number of recolouring steps required to transform the first colouring into the second is superpolynomial in the size of the graph: the minimum number of steps is Ω(2 N ), whereas the size of G N is O(N 2 ). This is in stark contrast to the k = 3 case, where it is known that the minimum number of recolouring steps is at most quadratic in the number of vertices. We also show that a class of bipartite graphs can be constructed with this property, and that: (i) for 4 ≤ k ≤ 6 planar graphs and (ii) for k = 4 bipartite planar graphs can be constructed with this property. This provides a remarkable correspondence between the tractability of the problem and its underlying structure.

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Section: Theorem 2 ([4]) Let G Be a 3-colourable Graph With N Verticsupporting

confidence: 67%

Finding Paths between graph colourings: PSPACE-completeness and superpolynomial distances

Bonsma

Cereceda

2009

Theoretical Computer Science

131

134

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“…In work by Gopalan et al [34], later corrected by Schwerdtfeger [130], an analogous dichotomy theorem was obtained for the reachability problem. By defining the class of tight relations, a superset of the Schaefer relations, the authors were able to show that reachability is in P for formulas built from tight relations and PSPACE-complete for all other formulas in the framework.…”

Section: Shortest Transformationmentioning

confidence: 82%

“…Typically an intractable source problem is NP-complete and an intractable reachability problem is PSPACE-complete, where the class PSPACE includes all problems that can be solved using polynomial space [12]. Polynomial-time reachability algorithms have been developed for tractable source problems such as 2-COLORING [31][32][33], MATCHING [4], MINIMUM SPANNING TREE [4], and 2-SATISFIABILITY [34]. Reachability has been shown to be PSPACE-complete for many NP-complete source problems, such as CLIQUE [4], 4-COLORING [35], INDEPENDENT SET [4,17], 3-SATISFIABILITY [34], VERTEX COVER [4], DOMINATING SET [4], LIST EDGE-COLORING [36], LIST L(2, 1)-LABELING [37], INTEGER PROGRAMMING [4], STEINER TREE [38], and SET COVER [4].…”

Section: Tools For Proving the Complexity Of Reachabilitymentioning

confidence: 99%

“…Given that SATISFIABILITY is NP-complete, it is not surprising to discover that reachability of SATISFIABILITY RECONFIGURATION is PSPACE-complete [34]. To determine the dividing line between tractable and intractable classes of formulas for reachability, researchers have built on Schaefer's dichotomy theorem [129] for the classical problem of SATISFIABILITY.…”

Section: Shortest Transformationmentioning

confidence: 99%

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Introduction to Reconfiguration

2018

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Reconfiguration is concerned with relationships among solutions to a problem instance, where the reconfiguration of one solution to another is a sequence of steps such that each step produces an intermediate feasible solution. The solution space can be represented as a reconfiguration graph, where two vertices representing solutions are adjacent if one can be formed from the other in a single step. Work in the area encompasses both structural questions (Is the reconfiguration graph connected?) and algorithmic ones (How can one find the shortest sequence of steps between two solutions?) This survey discusses techniques, results, and future directions in the area.

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“…The problem arises when we wish to find a step-by-step transformation between two feasible solutions of a problem such that all intermediate results are also feasible and each step abides by a fixed reconfiguration rule (i.e., an adjacency relation defined on feasible solutions of the original problem). This kind of reconfiguration problem has been studied extensively for several well-known problems, including independent set [2,5,7,10,11,13,15,19,[21][22][23], satisfiability [9,20], set cover, clique, matching [13], vertexcoloring [3,6,8,23], list edge-coloring [14,17], list L(2, 1)-labeling [16], subset sum [12], shortest path [4,18], and so on.…”

Section: Introductionmentioning

confidence: 99%

Polynomial-Time Algorithm for Sliding Tokens on Trees

Demaine¹,

Demaine²,

Fox-Epstein

et al. 2014

Algorithms and Computation

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Abstract. Suppose that we are given two independent sets I b and Ir of a graph such that |I b | = |Ir|, and imagine that a token is placed on each vertex in I b . Then, the sliding token problem is to determine whether there exists a sequence of independent sets which transforms I b into Ir so that each independent set in the sequence results from the previous one by sliding exactly one token along an edge in the graph. This problem is known to be PSPACE-complete even for planar graphs, and also for bounded treewidth graphs. In this paper, we thus study the problem restricted to trees, and give the following three results: (1) the decision problem is solvable in linear time; (2) for a yes-instance, we can find in quadratic time an actual sequence of independent sets between I b and Ir whose length (i.e., the number of token-slides) is quadratic; and (3) there exists an infinite family of instances on paths for which any sequence requires quadratic length.

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The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies

Cited by 135 publications

References 28 publications

Finding Paths between graph colourings: PSPACE-completeness and superpolynomial distances

Finding Paths between graph colourings: PSPACE-completeness and superpolynomial distances

Introduction to Reconfiguration

Polynomial-Time Algorithm for Sliding Tokens on Trees

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