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163
Parametric Integer Programming
 RAIRO Recherche Op'erationnelle
, 1988
"... When analysing computer programs (especially numerical programs in which arrays are used extensively), one is often confronted with integer programming problems. These problems have three peculiarities: ffl feasible points are ranked according to lexicographic order rather than the usual linear ec ..."
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Cited by 164 (19 self)
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When analysing computer programs (especially numerical programs in which arrays are used extensively), one is often confronted with integer programming problems. These problems have three peculiarities: ffl feasible points are ranked according to lexicographic order rather than the usual linear economic function; ffl the feasible set depends on integer parameters; ffl one is interested only in exact solutions. The difficulty is somewhat alleviated by the fact that problems sizes are usually quite small. In this paper we show that: ffl the classical simplex algorithm has no difficulty in handling lexicographic ordering; ffl the algorithm may be executed in symbolic mode, thus giving the solution of continuous parametric problems; ffl the method may be extended to problems in integers. We prove that the resulting algorithm always terminate and give an estimate of its complexity. R'esum'e L'analyse s'emantique des programmes (sp'ecialement des programmes num'eriques utilisant de...
On the Solution of Traveling Salesman Problems
 DOC. MATH. J. DMV
, 1998
"... Following the theoretical studies of J.B. Robinson and H.W. Kuhn in the late 1940s and the early 1950s, G.B. Dantzig, R. Fulkerson, and S.M. Johnson demonstrated in 1954 that large instances of the TSP could be solved by linear programming. Their approach remains the only known tool for solving TS ..."
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Cited by 164 (7 self)
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Following the theoretical studies of J.B. Robinson and H.W. Kuhn in the late 1940s and the early 1950s, G.B. Dantzig, R. Fulkerson, and S.M. Johnson demonstrated in 1954 that large instances of the TSP could be solved by linear programming. Their approach remains the only known tool for solving TSP instances with more than several hundred cities; over the years, it has evolved further through the work of M. Grötschel , S. Hong , M. Jünger , P. Miliotis , D. Naddef , M. Padberg
Edmonds polytopes and a hierarchy of combinatorial problems
, 2006
"... Let S be a set of linear inequalities that determine a bounded polyhedron P. The closure of S is the smallest set of inequalities that contains S and is closed under two operations: (i) taking linear combinations of inequalities, (ii) replacing an inequality Σaj xj ≤ a0, where a1,a2,...,an are integ ..."
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Cited by 146 (0 self)
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Let S be a set of linear inequalities that determine a bounded polyhedron P. The closure of S is the smallest set of inequalities that contains S and is closed under two operations: (i) taking linear combinations of inequalities, (ii) replacing an inequality Σaj xj ≤ a0, where a1,a2,...,an are integers, by the inequality Σaj xj ≤ a with a ≥[a0]. Obviously, if integers x1,x2,...,xn satisfy all the inequalities in S, then they satisfy also all inequalities in the closure of S. Conversely, let Σcj xj ≤ c0 hold for all choices of integers x1,x2,...,xn, that satisfy all the inequalities in S. Then we prove that Σcj xj ≤ c0 belongs to the closure of S. To each integer linear programming problem, we assign a nonnegative integer, called its rank. (The rank is the minimum number of iterations of the operation (ii) that are required in order to eliminate the integrality constraint.) We prove that there is no upper bound on the rank of problems arising from the search for largest independent sets in graphs.
Lower Bounds for Resolution and Cutting Plane Proofs and Monotone Computations
, 1997
"... We prove an exponential lower bound on the length of cutting plane proofs. The proof uses an extension of a lower bound for monotone circuits to circuits which compute with real numbers and use nondecreasing functions as gates. The latter result is of independent interest, since, in particular, i ..."
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Cited by 135 (5 self)
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We prove an exponential lower bound on the length of cutting plane proofs. The proof uses an extension of a lower bound for monotone circuits to circuits which compute with real numbers and use nondecreasing functions as gates. The latter result is of independent interest, since, in particular, it implies an exponential lower bound for some arithmetic circuits.
A Fast PseudoBoolean Constraint Solver
, 2003
"... Linear PseudoBoolean (LPB) constraints denote inequalities between arithmetic sums of weighted Boolean functions and provide a significant extension of the modeling power of purely propositional constraints. They can be used to compactly describe many discrete EDA problems with constraints on linea ..."
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Cited by 101 (1 self)
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Linear PseudoBoolean (LPB) constraints denote inequalities between arithmetic sums of weighted Boolean functions and provide a significant extension of the modeling power of purely propositional constraints. They can be used to compactly describe many discrete EDA problems with constraints on linearly combined, parameterized weights, yet also offer efficient search strategies for proving or disproving whether a satisfying solution exists. Furthermore, corresponding decision procedures can easily be extended for minimizing or maximizing an LPB objective function, thus providing a core optimization method for many problems in logic and physical synthesis. In this paper we review how recent advances in satisfiability (SAT) search can be extended for pseudoBoolean constraints and describe a new LPB solver that is based on generalized constraint propagation and conflictbased learning. We present a comparison with other, stateoftheart LPB solvers which demonstrates the overall efficiency of our method.
Lower Bounds for Cutting Planes Proofs with Small Coefficients
, 1995
"... We consider smallweight Cutting Planes (CP ) proofs; that is, Cutting Planes (CP ) proofs with coefficients up to P oly(n). We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP proofs, for a family of tautologies based on the cl ..."
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Cited by 77 (19 self)
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We consider smallweight Cutting Planes (CP ) proofs; that is, Cutting Planes (CP ) proofs with coefficients up to P oly(n). We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP proofs, for a family of tautologies based on the clique function. Because Resolution is a special case of smallweight CP , our method also gives a new and simpler exponential lower bound for Resolution. We also prove the following two theorems : (1) Treelike CP proofs cannot polynomially simulate nontreelike CP proofs. (2) Treelike CP proofs and BoundeddepthFrege proofs cannot polynomially simulate each other. Our proofs also work for some generalizations of the CP proof system. In particular, they work for CP with a deduction rule, and also for proof systems that allow any formula with small communication complexity, and any set of sound rules of inference. 1 Introduction One of the most fundamental questions in pro...
CHVATAL CLOSURES FOR MIXED INTEGER PROGRAMMING PROBLEMS
, 1990
"... Chvátal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality. This viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems. T ..."
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Cited by 65 (0 self)
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Chvátal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality. This viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems. The basic ingredient in these cuttingplane proofs is that for a polyhedron P and integral vector w, if max(wx]x ~ P, wx integer} = t, then wx ~ t is valid for all integral vectors in P. We consider the variant of this step where the requirement that wx be integer may be replaced by the requirement that #x be integer for some other integral vector #. The cuttingplane proofs thus obtained may be seen either as an abstraction of Gomory's mixed integer cuttingplane technique or as a proof version of a simple class of the disjunctive cutting planes studied by Balas and Jeroslow. Our main result is that for a given polyhedron P, the set of vectors that satisfy every cutting plane for P with respect to a specified subset of integer variables is again a polyhedron. This allows us to obtain a finite recursive procedure for generating the mixed integer hull of a polyhedron, analogous to the process of repeatedly taking Chvátal closures in the integer programming case. These results are illustrated with a number of examples from combinatorial optimization. Our work can be seen as a continuation of that of Nemhauser and Wolsey on mixed integer cutting planes.
SCIP: solving constraint integer programs
, 2009
"... Constraint integer programming (CIP) is a novel paradigm which integrates constraint programming (CP), mixed integer programming (MIP), and satisfiability (SAT) modeling and solving techniques. In this paper we discuss the software framework and solver SCIP (Solving Constraint Integer Programs), wh ..."
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Cited by 53 (0 self)
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Constraint integer programming (CIP) is a novel paradigm which integrates constraint programming (CP), mixed integer programming (MIP), and satisfiability (SAT) modeling and solving techniques. In this paper we discuss the software framework and solver SCIP (Solving Constraint Integer Programs), which is free for academic and noncommercial use and can be downloaded in source code. This paper gives an overview of the main design concepts of SCIP and how it can be used to solve constraint integer programs. To illustrate the performance and flexibility of SCIP, we apply it to two different problem classes. First, we consider mixed integer programming and show by computational experiments that SCIP is almost competitive to specialized commercial MIP solvers, even though SCIP supports the more general constraint integer programming paradigm. We develop new ingredients that improve current MIP solving technology. As a second application, we employ SCIP to solve chip design verification problems as they arise in the logic design of integrated circuits. This application goes far beyond traditional MIP solving, as it includes several highly nonlinear constraints, which can be handled nicely within the constraint integer programming framework. We show anecdotally how the different solving techniques from MIP, CP, and SAT work together inside SCIP to deal with such constraint classes. Finally, experimental results show that our approach outperforms current stateoftheart techniques for proving the validity of properties on circuits containing arithmetic.
Automatic Parallelization in the Polytope Model
 Laboratoire PRiSM, Université des Versailles StQuentin en Yvelines, 45, avenue des ÉtatsUnis, F78035 Versailles Cedex
, 1996
"... . The aim of this paper is to explain the importance of polytope and polyhedra in automatic parallelization. We show that the semantics of parallel programs is best described geometrically, as properties of sets of integral points in ndimensional spaces, where n is related to the maximum nesting de ..."
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Cited by 48 (3 self)
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. The aim of this paper is to explain the importance of polytope and polyhedra in automatic parallelization. We show that the semantics of parallel programs is best described geometrically, as properties of sets of integral points in ndimensional spaces, where n is related to the maximum nesting depth of DO loops. The needed properties translate nicely to properties of polyhedra, for which many algorithms have been designed for the needs of optimization and operation research. We show how these ideas apply to scheduling, placement and parallel code generation. R'esum'e Le but de cet article est d'expliquer le role jou'e par les poly`edres et les polytopes en parall'elisation automatique. On montre que la s'emantique d'un programme parall`ele se d'ecrit naturellement sous forme g'eom'etrique, les propri'et'es du programme 'etant formalis'ees comme des propri'et'es d'ensemble de points dans un espace `a n dimensions. n est li'e `a la profondeur maximale d'imbrication des boucles DO. Les...
Mixed Integer Programming Methods for Computing Nonmonotonic Deductive Databases
, 1994
"... Though the declarative semantics of both explicit and nonmonotonic negation in logic programs has been studied extensively, relatively little work has been done on computation and implementation of these semantics. In this paper, we study three different approaches to computing stable models of logi ..."
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Cited by 45 (8 self)
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Though the declarative semantics of both explicit and nonmonotonic negation in logic programs has been studied extensively, relatively little work has been done on computation and implementation of these semantics. In this paper, we study three different approaches to computing stable models of logic programs based on mixed integer linear programming methods for automated deduction introduced by R. Jeroslow. We subsequently discuss the relative efficiency of these algorithms. The results of experiments with a prototype compiler implemented by us tend to confirm our theoretical discussion. In contrast to resolution, the mixed integer programming methodology is both fully declarative and handles reuse of old computations gracefully. We also introduce, compare, implement, and experiment with linear constraints corresponding to four semantics for "explicit" negation in logic programs: the fourvalued annotated semantics [3], the GelfondLifschitz semantics [12], the overdetermined models ...