A Computational Trichotomy for Connectivity of Boolean Satisfiability

Schwerdtfeger, Konrad W.

doi:10.3233/sat190097

Cited by 13 publications

(28 citation statements)

References 18 publications

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“…This was slightly corrected (with a further correction in 2015) and extended to several similar problems by Schwerdtfeger [Sch14; Sch16], while a trichotomy was shown for the problem of finding shortest paths by Mouawad et al [Mou+15]. Both [Gop+09] and [Sch14] asked whether their results could be extended to larger domains. Our work can be seen as a step in this direction, but limited to only one symmetric relation of arity 2.…”

Section: Figurementioning

confidence: 97%

Homomorphism Reconfiguration via Homotopy

Wrochna¹

2020

SIAM J. Discrete Math.

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For a fixed graph H, we consider the H-Recoloring problem : given a graph G and two H-colorings of G, i.e., homomorphisms from G to H, can one be transformed into the other by changing one color at a time, maintaining an H-coloring throughout. This is the same as finding a path in the Hom(G, H) complex. For H = K k this is the problem of finding paths between k-colorings, which was recently shown to be in P for k ≤ 3 and PSPACE-complete otherwise. We generalize the positive side of this dichotomy by providing an algorithm that solves the problem in polynomial time for any H with no C 4 subgraph. This gives a large class of constraints for which finding solutions to the Constraint Satisfaction Problem is NP-complete, but finding paths in the solution space is in P.The algorithm uses a characterization of possible reconfiguration sequences (paths in Hom(G, H)), whose main part is a purely topological condition described in terms of the fundamental groupoid of H seen as a topological space.

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Section: Figurementioning

confidence: 97%

Homomorphism Reconfiguration via Homotopy

Wrochna¹

2020

SIAM J. Discrete Math.

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“…In particular, our work was inspired by the paper of Gopalan, Kolaitis, Maneva, and Papadimitriou [GKMP06], which shows that determining whether two solutions x and y of a Boolean formula are connected through the solution space is either in P or is PSPACE-complete, depending on the constraint types allowed in the formula. (Note: A minor error in Reference [GKMP06] was recently corrected in the work of Schwerdtfeger [Sch13].) More recently, Mouawad, Nishimura, Pathak and Raman [MNPR14] studied the variant of this problem in which one seeks the shortest possible Boolean reconfiguration path; they show this problem is either in P, NP-complete, or PSPACE-complete.…”

Section: Introductionmentioning

confidence: 99%

Ground State Connectivity of Local Hamiltonians

Gharibian

Sikora

2015

Automata, Languages, and Programming

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In this work we consider the ground space connectivity problem for commuting local Hamiltonians. The ground space connectivity problem asks whether it is possible to go from one (efficiently preparable) state to another by applying a polynomial length sequence of 2-qubit unitaries while remaining at all times in a state with low energy for a given Hamiltonian H. It was shown in [GS15] that this problem is QCMA-complete for general local Hamiltonians, where QCMA is defined as QMA with a classical witness and BQP verifier. Here we show that the commuting version of the problem is also QCMA-complete. This provides one of the first examples where commuting local Hamiltonians exhibit complexity theoretic hardness equivalent to general local Hamiltonians.

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“…Combining this result with result (ii), we obtain a complexity dichotomy for the reconfiguration problem of SAT in terms of the complexity index. We compare this dichotomy result with the dichotomy result in [8,22] in the next subsection.…”

Section: Main Results Of the Papermentioning

confidence: 99%

Trichotomy for the Reconfiguration Problem of Integer Linear Systems

Kimura

Suzuki

2020

WALCOM: Algorithms and Computation

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In this paper, we consider the reconfiguration problem of integer linear systems. In this problem, we are given an integer linear system I and two feasible solutions s and t of I, and then asked to transform s to t by changing a value of only one variable at a time, while maintaining a feasible solution of I throughout. Z(I) for I is the complexity index introduced by Kimura and Makino (Discrete Applied Mathematics 200:67-78, 2016), which is defined by the sign pattern of the input matrix. We analyze the complexity of the reconfiguration problem of integer linear systems based on the complexity index Z(I) of given I. We then show that the problem is (i) solvable in constant time if Z(I) is less than one, (ii) weakly coNP-complete and pseudo-polynomially solvable if Z(I) is exactly one, and (iii) PSPACE-complete if Z(I) is greater than one. Since the complexity indices of Horn and two-variable-par-inequality integer linear systems are at most one, our results imply that the reconfiguration of these systems are in coNP and pseudo-polynomially solvable. Moreover, this is the first result that reveals coNP-completeness for a reconfiguration problem, to the best of our knowledge.

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A Computational Trichotomy for Connectivity of Boolean Satisfiability

Cited by 13 publications

References 18 publications

Homomorphism Reconfiguration via Homotopy

Homomorphism Reconfiguration via Homotopy

Ground State Connectivity of Local Hamiltonians

Trichotomy for the Reconfiguration Problem of Integer Linear Systems

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