Optimal Length Resolution Refutations of Difference Constraint Systems

Subramani, K.

doi:10.1007/s10817-009-9139-4

Cited by 31 publications

(20 citation statements)

References 51 publications

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“…We already know that if a DCS is unsatisfiable, then there must exist a simple negative cost cycle in the corresponding constraint network [Sub09]. The same applies for a WDCS.…”

Section: The Wolrr Problem (Add Rule)mentioning

confidence: 91%

“…A lot of the work in finding short refutations focuses on discrete domains. However, [Sub09] considers the problem of finding optimal length refutations for systems of difference constraints. In that paper it was shown that short refutations exist for difference constraints and also that the optimal length refutations for systems of difference constraints can be determined in polynomial time.…”

Section: Constraint Systemsmentioning

confidence: 99%

“…In that paper it was shown that short refutations exist for difference constraints and also that the optimal length refutations for systems of difference constraints can be determined in polynomial time. The algorithm in [Sub09] is based runs in time O(n 3 • log n) on a DCS with n variables and utilizes dynamic programming. In [SWG13], a time of O(m • n • k), where m is the number of constraints and k is the length of the shortest refutation was obtained utilizing a different dynamic programming technique.…”

Section: Constraint Systemsmentioning

confidence: 99%

“…It is worth noting that in DCSs, linear and integer feasibility coincide and therefore, the departure is not strict. Furthermore, as pointed out in [Sub09], every minimal refutation (i.e., a refutation without redundant constraints) is necessarily read-once and literal-once, since every minimal refutation corresponds to a simple negative cost cycle in the corresponding constraint network [CLRS01]. In this paper though, we consider the problem of read-once refutations in UCSs.…”

Section: Constraint Systemsmentioning

confidence: 99%

“…In [Sub04b], optimal-length tree-like proofs were studied for 2CNF formulae. In [Sub09], it was established that read-once, tree-like, and dag-like proofs coincide for difference constraints systems.…”

Section: An Fptas For Wolrrmentioning

confidence: 99%

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Analyzing Satisfiability and Refutability in Selected Constraint Systems

Wojciechowski¹

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This dissertation is concerned with the satisfiability and refutability problems for several constraint systems. We examine both boolean constraint systems, in which each variable is limited to the values true and false, and polyhedral constraint systems, in which each variable is limited to the set of real numbers R in the case of linear polyhedral systems or the set of integers Z in the case of integer polyhedral systems. An important aspect of our research is that we focus on providing certificates. That is, we provide satisfying assignments or easily checkable proofs of infeasibility depending on whether the instance is feasible or not. Providing easily checkable certificates has become a much sought after feature in algorithms, especially in light of spectacular failures in the implementations of some well-known algorithms. There exist a number of problems in the constraint-solving domain for which efficient algorithms have been proposed, but which lack a certifying counterpart. When examining boolean constraint systems, we specifically look at systems of 2-CNF clauses and systems of Horn clauses. When examining polyhedral constraint systems, we specifically look at systems of difference constraints, systems of UTVPI constraints, and systems of Horn constraints. For each examined system, we determine several properties of general refutations and determine the complexity of finding restricted refutations. These restricted forms of refutation include read-once refutations, in which each constraint can be used at most once; literal-once refutations, in which for each literal at most one constraint containing that literal can be used; and unit refutations, in which each step of the refutation must use a constraint containing exactly one literal. The advantage of read-once refutations is that they are guaranteed to be short. Thus, while not every constraint system has a read-once refutation, the small size of the refutation guarantees easy checkability. we have collaborated several times. He has contributed greatly to my understanding of horn constraints and this understanding has made many of the results in this dissertation possible. West Virginia University has provided me with the education and reseach opportunities that made this work possible. In particular the Lane Department of Computer Science and Electrical Engineering, along with its chair Dr. Brian Woerner, have given me the knowledge and funding to be able to perform the research which resulted in this dissertation.

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“…We already know that if a DCS is unsatisfiable, then there must exist a simple negative cost cycle in the corresponding constraint network [Sub09]. The same applies for a WDCS.…”

Section: The Wolrr Problem (Add Rule)mentioning

confidence: 91%

Section: Constraint Systemsmentioning

confidence: 99%