Shortest Reconfiguration Paths in the Solution Space of Boolean Formulas

Mouawad, Amer E.; Nishimura, Nobuya; Pathak, Vinayak; Raman, Venkatesh

doi:10.1137/16m1065288

Cited by 23 publications

(13 citation statements)

References 26 publications

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“…From the viewpoint of theoretical computer science, one of the most important problems is the 3SAT problem. For this 3SAT problem, a similar trichotomy for the complexity of finding a shortest sequence has been shown recently; that is, for the reconfiguration problem of 3SAT, finding a shortest sequence between two satisfiable assignments is in P, NP-complete, or PSPACE-complete in certain conditions [19]. In general, the reconfiguration problems tend to be PSPACEcomplete, and some polynomial time algorithms are shown in restricted cases.…”

Section: Introductionmentioning

confidence: 70%

Shortest Reconfiguration of Sliding Tokens on a Caterpillar

Uehara

2016

WALCOM: Algorithms and Computation

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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. The sliding token problem is one of the reconfiguration problems that attract the attention from the viewpoint of theoretical computer science. The reconfiguration problems tend to be PSPACE-complete in general, and some polynomial time algorithms are shown in restricted cases. Recently, the problems that aim at finding a shortest reconfiguration sequence are investigated. For the 3SAT problem, a trichotomy for the complexity of finding the shortest sequence has been shown; that is, it is in P, NPcomplete, or PSPACE-complete in certain conditions. In general, even if it is polynomial time solvable to decide whether two instances are reconfigured with each other, it can be NP-complete to find a shortest sequence between them. Namely, finding a shortest sequence between two independent sets can be more difficult than the decision problem of reconfigurability between them. In this paper, we show that the problem for finding a shortest sequence between two independent sets is polynomial time solvable for some graph classes which are subclasses of the class of interval graphs. More precisely, we can find a shortest sequence between two independent sets on a graph G in polynomial time if either G is a proper interval graph, a trivially perfect graph, or a caterpillar. As far as the authors know, this is the first polynomial time algorithm for the shortest sliding token problem for a graph class that requires detours.Recently, the reconfiguration problems attract the attention from the viewpoint of theoretical computer science. The problem arises when we wish to find a stepby-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, that is, an adjacency relation defined on feasible solutions of the original problem. The reconfiguration problems have been studied extensively for several

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Section: Introductionmentioning

confidence: 70%

Shortest Reconfiguration of Sliding Tokens on a Caterpillar

Uehara

2016

WALCOM: Algorithms and Computation

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“…Such problems (also referred to as the s − t-connectivity problems for Boolean satisfiability) have been considered extensively before [9,22,23,29,26,28]. Here we investigate the complexity of the reconfiguration versions of Boolean satisfiability problems in which the variable-clause incidence graph is planar.…”

Section: Planar Nae 3-sat Reconfigurationmentioning

confidence: 99%

Reconfiguration of satisfying assignments and subset sums: Easy to find, hard to connect

Cardinal¹,

Demaine²,

Eppstein³

et al. 2020

Theoretical Computer Science

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We consider the computational complexity of reconfiguration problems, in which one is given two combinatorial configurations satisfying some constraints, and is asked to transform one into the other using elementary transformations, while satisfying the constraints at all times. Such problems appear naturally in many contexts, such as model checking, motion planning, enumeration and sampling, and recreational mathematics. We provide hardness results for problems in this family, in which the constraints and operations are particularly simple.More precisely, we prove the PSPACE-completeness of the following decision problems:• Given two satisfying assignments to a planar monotone instance of Not-All-Equal 3-SAT, can one assignment be transformed into the other by single variable "flips" (assignment changes), preserving satisfiability at every step?• Given two subsets of a set S of integers with the same sum, can one subset be transformed into the other by adding or removing at most three elements of S at a time, such that the intermediate subsets also have the same sum?• Given two points in {0, 1} n contained in a polytope P specified by a constant number of linear inequalities, is there a path in the n-hypercube connecting the two points and contained in P ?These problems can be interpreted as reconfiguration analogues of standard problems in NP. Interestingly, the instances of the NP problems that appear as input to the reconfiguration problems in our reductions can be shown to lie in P. In particular, the elements of S and the coefficients of the inequalities defining P can be restricted to have logarithmic bit-length.

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“…Analogously, the k-recolouring problem also generalizes to reconfiguration problems for H-colourings and CSP, both of which are well studied; see, e.g., [4-6, 13, 20] and [2,8,10,11,[14][15][16]18], respectively. In particular, Gopalan et al [10] proved a dichotomy theorem for the reconfiguration variation of CSP(H) for structures H with two vertices.…”

Section: Introductionmentioning

confidence: 99%

Recolouring Homomorphisms to triangle-free reflexive graphs

Lee¹,

Noel²,

Siggers³

2021

Preprint

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For a graph H, the H-recolouring problem Recol(H) asks, for two given homomorphisms from a given graph G to H, if one can get between them by a sequence of homomorphisms of G to H in which consecutive homomorphisms differ on only one vertex. We show that, if G and H are reflexive and H is triangle-free, then this problem can be solved in polynomial time. This shows, at the same time, that the closely related H-reconfiguration problem Recon(H) of deciding whether two given homomorphisms from a given graph G to H are in the same component of the Hom-graph Hom(G, H), can be solved in polynomial time for triangle-free reflexive graphs H.

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Shortest Reconfiguration Paths in the Solution Space of Boolean Formulas

Cited by 23 publications

References 26 publications

Shortest Reconfiguration of Sliding Tokens on a Caterpillar

Shortest Reconfiguration of Sliding Tokens on a Caterpillar

Reconfiguration of satisfying assignments and subset sums: Easy to find, hard to connect

Recolouring Homomorphisms to triangle-free reflexive graphs

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