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Using CSP LookBack Techniques to Solve Exceptionally Hard SAT Instances
 Principles and Practice of Constraint Programming
, 1996
"... Abstract. While CNF propositional satisfiability (SAT) is a subclass of the more general constraint satisfaction problem (CSP), conventional wisdom has it that some wellknown CSP lookback techniques including backjumping and learning are of little use for SAT. We enhance the Tableau SAT algor ..."
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Cited by 34 (1 self)
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Abstract. While CNF propositional satisfiability (SAT) is a subclass of the more general constraint satisfaction problem (CSP), conventional wisdom has it that some wellknown CSP lookback techniques including backjumping and learning are of little use for SAT. We enhance the Tableau SAT algorithm of Crawford and Auton with lookback techniques and evaluate its performance on problems specifically designed to challenge it. The Random 3SAT problem space has commonly been used to benchmark SAT algorithms because consistently difficult instances can be found near a region known as the phase transition. We modify Random 3SAT in two ways which make instances even harder. First, we evaluate problems with structural regularities and find that CSP lookback techniques offer little advantage. Second, we evaluate problems in which a hard unsatisfiable instance of medium size is embedded in a larger instance, and we find the lookback enhancements to be indispensable. Without them, most instances are “exceptionally hard ”orders of magnitude harder than typical Random 3SAT instances with the same surface characteristics.
Combinatorial Benders’ Cuts for MixedInteger Linear Programming
 Operations Research
"... MixedInteger Programs (MIP’s) involving logical implications modelled through bigM coefficients, are notoriously among the hardest to solve. In this paper we propose and analyze computationally an automatic problem reformulation of quite general applicability, aimed at removing the model dependenc ..."
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Cited by 8 (0 self)
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MixedInteger Programs (MIP’s) involving logical implications modelled through bigM coefficients, are notoriously among the hardest to solve. In this paper we propose and analyze computationally an automatic problem reformulation of quite general applicability, aimed at removing the model dependency on the bigM coefficients. Our solution scheme defines a master Integer Linear Problem (ILP) with no continuous variables, which contains combinatorial information on the feasible integer variable combinations that can be “distilled ” from the original MIP model. The master solutions are sent to a slave Linear Program (LP), which validates them and possibly returns combinatorial inequalities to be added to the current master ILP. The inequalities are associated to minimal (or irreducible) infeasible subsystems of a certain linear system, and can be separated efficiently in case the master solution is integer. The overall solution mechanism resembles closely the Benders ’ one, but the cuts we produce are purely combinatorial and do not depend on the bigM values used in the MIP formulation. This produces an LP relaxation of the master problem which can be considerably tighter than the one associated with original MIP formulation. Computational results on two specific classes of hardtosolve MIP’s indicate the new method produces a reformulation which can be solved some orders of magnitude faster than the original MIP model.
Resolution Search and Dynamic BranchandBound
, 2000
"... Abstract __ _ A novel approach to pure 01 integer programming problems called Resolution Search has been proposed by Chvatal (l997) as an alternative to implicit enumeration, with a demonstration that the method can yield more effective branching strategies. We show that an earlier method called Dy ..."
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Cited by 2 (0 self)
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Abstract __ _ A novel approach to pure 01 integer programming problems called Resolution Search has been proposed by Chvatal (l997) as an alternative to implicit enumeration, with a demonstration that the method can yield more effective branching strategies. We show that an earlier method called Dynamic BranchandBound (B&B) yields the same branching strategies as Resolution Search, and other strategic alternatives in addition. Moreover, Dynamic B&B is not restricted to pure 01 problems, but applies to general mixed integer programs containing both general integer and continuous variables. We provide examples comparing Resolution Search to enhanced variants. We also show the relation of these approaches to Dynamic B&B, suggesting the value of further study of this neglected approach.
Orthogonalization of a Boolean Function
, 2000
"... Orthogonalization is the process of transforming a conjunctive normal form of a Boolean function to orthogonal conjunctive normal form, i.e. to a normal form in which any two clauses contain a pair of complementary literals. Orthogonal disjunctive normal form is defined similarly. The problem is of ..."
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Orthogonalization is the process of transforming a conjunctive normal form of a Boolean function to orthogonal conjunctive normal form, i.e. to a normal form in which any two clauses contain a pair of complementary literals. Orthogonal disjunctive normal form is defined similarly. The problem is of great relevance in several application, e.g. the reliability theory and the propositional satisfiability problem. We propose a procedure for transforming an arbitrary CNF or DNF to an orthogonal one, and present the results of computational experiments carried out on randomly generated Boolean formulae.
EXPLOITING STRUCTURE IN INTEGER PROGRAMS
, 2011
"... This dissertation argues the case for exploiting certain structures in integer linear programs. Integer linear programming is a wellknown optimisation problem, which seeks the optimum of a linear function of variables, whose values are required to be integral as well as to satisfy certain linear eq ..."
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This dissertation argues the case for exploiting certain structures in integer linear programs. Integer linear programming is a wellknown optimisation problem, which seeks the optimum of a linear function of variables, whose values are required to be integral as well as to satisfy certain linear equalities and inequalities. The state of the art in solvers for this problem is the “branch and bound ” approach. The performance of such solvers depends crucially on four types of inbuilt heuristics: primal, improvement, branching, and cutseparation or, more generally, bounding heuristics. Such heuristics in generalpurpose solvers have not, until recently, exploited structure in integer linear programs beyond the recognition of certain types of singlerow constraints. Many alternative approaches to integer linear programming can be cast in the following, novel framework. “Structure” in any integer linear program
Solving the 0–1 Multidimensional Knapsack Problem with Resolution Search
, 905
"... Abstract. We propose an exact method which combines the resolution search and branch & bound algorithms for solving the 0–1 Multidimensional Knapsack Problem. This algorithm is able to prove large–scale strong correlated instances. The optimal values of the 10 constraint, 500 variable instances of t ..."
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Abstract. We propose an exact method which combines the resolution search and branch & bound algorithms for solving the 0–1 Multidimensional Knapsack Problem. This algorithm is able to prove large–scale strong correlated instances. The optimal values of the 10 constraint, 500 variable instances of the ORLibrary are exposed. These values were previously unknown. 1
Author manuscript, published in "VI ALIO/EURO Workshop on Applied Combinatorial Optimization, Buenos Aires: Argentina (2008)" Solving the 0–1 Multidimensional Knapsack Problem with Resolution Search
, 2009
"... Abstract. We propose an exact method which combines the resolution search and branch & bound algorithms for solving the 0–1 Multidimensional Knapsack Problem. This algorithm is able to prove large–scale strong correlated instances. The optimal values of the 10 constraint, 500 variable instances of t ..."
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Abstract. We propose an exact method which combines the resolution search and branch & bound algorithms for solving the 0–1 Multidimensional Knapsack Problem. This algorithm is able to prove large–scale strong correlated instances. The optimal values of the 10 constraint, 500 variable instances of the ORLibrary are exposed. These values were previously unknown. 1