Shortest Reconfiguration Paths in the Solution Space of Boolean Formulas

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

doi:10.1007/978-3-662-47672-7_80

Cited by 20 publications

(16 citation statements)

References 29 publications

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“…Mouawad, Nishimura, Pathak, and Raman [133] considered shortest transformation, and obtained a trichotomy, based on both tight relations and navigable relations, a subset of tight relations. They showed that the problem is in P for formulas built from navigable relations, NP-complete for formulas built from relations that are tight but not navigable, and PSPACE-complete otherwise.…”

Section: Shortest Transformationmentioning

confidence: 99%

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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Section: Shortest Transformationmentioning

confidence: 99%

Introduction to Reconfiguration

2018

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“…Generally speaking, finding shortest transformations is much more difficult if we need a detour, which touches an element that is not in the symmetric difference of the source and target configurations. For such a detour-required case, only a few polynomial-time algorithms are known for shortest variants, e.g., satisfying assignments of a certain Boolean formulas by Mouawad et al [18], and independent sets under the TS operation for caterpillars by Yamada and Uehara [24]. Note that Matching Reconfiguration belongs to the detourrequired case; recall the example in Figure 1, where we need to use the edge in…”

Section: Exact Matching Diametermentioning

confidence: 99%

“…Recently, there has been considerable interest in the complexity of finding shortest transformations between configurations. Examples include finding a shortest transformation between triangulations of planar point sets [21] and simple polygons [1], configurations of the Rubik's cube [6], and satisfying assignments of Boolean formulas [17]. For all of these problems, except the last one, we can decide efficiently if a transformation between two given configurations exists.…”

Section: Introductionmentioning

confidence: 99%

Shortest Reconfiguration of Matchings

Bousquet

Hatanaka

Ito

et al. 2019

Graph-Theoretic Concepts in Computer Science

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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

Cardinal

Demaine

Eppstein

et al. 2018

Lecture Notes in 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.extension of Schaefer's dichotomy to the connectivity of Boolean satisfiability due to Gopalan et al. [9], and an overview of the complexity of reconfiguration problems by Ito et al. [16]. In the canonical example of Boolean satisfiability, one is given two satisfying assignments to connect by a sequence of variable assignment flips, such that the formula remains satisfied at every step. The study of this type of question also benefits from the interest of puzzle designers and recreational mathematicians; token-sliding problems, for instance, are related to the famous 15-puzzle, popular in the late 19th century [30]. Combinatorial reconfiguration now constitutes a quickly developing field with dedicated research groups and workshops (such as the combinatorial reconfiguration workshop held in Banff in January 2017). For a more thorough survey and history of this family of problems, we refer to van den Heuvel [13].Reconfiguration of independent sets in graphs is among the most studied problem in this vein (see [4,6,8,14] for recent results) and is relevant to our findings. In these problems, one is given a graph G and two independent sets of G of the same size k, and the goal is to transform one into the other using elementary operations, preserving independence at every step. The operations consist either of "token slides", in which a vertex in the independent set is replaced by one of its neighbors...

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Shortest Reconfiguration Paths in the Solution Space of Boolean Formulas

Cited by 20 publications

References 29 publications

Introduction to Reconfiguration

Introduction to Reconfiguration

Shortest Reconfiguration of Matchings

Reconfiguration of Satisfying Assignments and Subset Sums: Easy to Find, Hard to Connect

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