Combining Logic and Optimization in Cutting Plane Theory

Bockmayr, Alexander; Eisenbrand, Friedrich

doi:10.1007/10720084_1

Cited by 7 publications

(3 citation statements)

References 31 publications

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“…However, FM elimination is not a valid proof system for a system of integer inequalities; [55] provides examples where FM is not sound. Proofs in integer arithmetic usually use cutting plane theory [9]. The resolution technique for clausal formulas has a number of variants including tree-like resolution, dag-like resolution, read-once resolution, linear resolution, and so on [13,42].…”

Section: Motivation and Related Workmentioning

confidence: 99%

Optimal Length Resolution Refutations of Difference Constraint Systems

Subramani

2009

J Autom Reasoning

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This paper is concerned with determining the optimal length resolution refutation (OLRR) of a system of difference constraints over an integral domain. The problem of finding short explanations for unsatisfiable difference constraint systems (DCS) finds applications in a number of design domains including program verification, proof theory, real-time scheduling and operations research. It is wellknown that resolution refutation is a sound and complete procedure to establish the unsatisfiability of boolean formulas in clausal form. The literature has also established that a variant of the resolution procedure called Fourier-Motzkin elimination is a sound and complete procedure for establishing the unsatisfiability of linear constraint systems (LCS). Our work in this paper first establishes that every DCS has a short (polynomial in the size of the input) resolution refutation and then shows that there exists a polynomial time algorithm to compute the optimal size refutation. One of the consequences of our work is that the Minimum Unsatisfiable Subset (MUS) of a DCS can be computed in polynomial time; computing the MUS of an unsatisfiable constraint set is an extremely important aspect of certifying algorithms.

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Section: Motivation and Related Workmentioning

confidence: 99%

Optimal Length Resolution Refutations of Difference Constraint Systems

Subramani

2009

J Autom Reasoning

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“…Some of the concepts that are most relevant to the work presented here are: decomposition approaches (e.g. Benders [3]) that solve parts of the problem with different techniques [10,14,19,21,24,33]; allowing different models/solvers to exchange information [32]; using linear programming to reduce the domains of variables or to fix them to certain values [4,11,32]; automatic reformulation of global constraints as systems of linear inequalities [30]; continuous relaxations of global constraints and disjunctions of linear systems [1,14,18,22,28,36,37,38]; understanding the generation of cutting planes as a form of logical inference [6,7]; strengthening the problem formulation by embedding the generation of valid cutting planes into CP constraints [12]; maintaining the continuous relaxation of a constraint updated when the domains of its variables change [29]; and using global constraints as a key component in the intersection of CP and OR [27].…”

Section: Previous Workmentioning

confidence: 99%

SIMPL: A System for Integrating Optimization Techniques

Aron

Hooker

Yunes

2004

Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems

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Abstract. In recent years, the Constraint Programming (CP) and Operations Research (OR) communities have explored the advantages of combining CP and OR techniques to formulate and solve combinatorial optimization problems. These advantages include a more versatile modeling framework and the ability to combine complementary strengths of the two solution technologies. This research has reached a stage at which further development would benefit from a general-purpose modeling and solution system. We introduce here a system for integrated modeling and solution called SIMPL. Our approach is to view CP and OR techniques as special cases of a single method rather than as separate methods to be combined. This overarching method consists of an infer-relax-restrict cycle in which CP and OR techniques may interact at any stage. We describe the main features of SIMPL and illustrate its usage with examples.

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“…Note that under the assumption that NP = co-NP, 3SAT formulas cannot have polynomial sized refutations. Exponential lower bounds for propositional formulas using cutting plane proof systems were first shown in Pudlák [1997]; applications of cutting plane theory to propositional proof systems are also discussed in Bockmayr and Eisnbrand [2000]. Exhaustive surveys of Propositional proof complexity can be found in Beame and Pitassi [1998] and Urquhart [1995].…”

Section: Introductionmentioning

confidence: 99%