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14
Branch and infer: a unifying framework for integer and finite domain constraint programming
 INFORMS J. Comput
, 1998
"... We introduce branch and infer, a unifying framework for integer linear programming and finite domain constraint programming. We use this framework to compare the two approaches with respect to their modeling and solving capabilities, to introduce symbolic constraint abstractions into integer program ..."
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Cited by 28 (2 self)
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We introduce branch and infer, a unifying framework for integer linear programming and finite domain constraint programming. We use this framework to compare the two approaches with respect to their modeling and solving capabilities, to introduce symbolic constraint abstractions into integer programming, and to discuss possible combinations of the two approaches. C ombinatorial problems are ubiquitous in many real world applications like scheduling, planning, transportation, assignment, and many others. Besides special purpose algorithms to compute exact or approximate solutions, there exist also general approaches to solve this kind of problem. We are interested here in two such approaches: • Integer linear programming (ILP) • Finite domain constraint programming (CP(FD)) Integer linear programming has a long tradition in operations research and has produced a large number of impressive results during the last 40 years, see for example [37, 30]. Finite domain constraint programming is a promising new approach for solving complex combinatorial problems, which combines recent progress in programming language design, like constraint logic programming[29] or concurrent constraint programming,[42] with efficient constraint solving techniques from mathematics, artificial intelligence, and operations research, see for example [49, 50]. The aim of this paper is to develop a unifying framework for integer linear programming and finite domain constraint programming. On the one hand, we want to clarify the relationship between these two approaches and identify (some of) their strengths and weaknesses. On the other hand, we want to show how each of the two approaches may profit from the other and indicate possible ways towards their integration. This continues our previous work in
Capacitated facility location: separation algorithms and computational experience
 Mathematical Programming
, 1998
"... We consider the polyhedral approach to solving the capacitated facility location problem. The valid inequalities considered are the knapsack, flow cover, effective capacity, single depot, and combinatorial inequalities. The flow cover, effective capacity, and single depot inequalities form subfamili ..."
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Cited by 23 (2 self)
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We consider the polyhedral approach to solving the capacitated facility location problem. The valid inequalities considered are the knapsack, flow cover, effective capacity, single depot, and combinatorial inequalities. The flow cover, effective capacity, and single depot inequalities form subfamilies of the general family of submodular inequalities. The separation problem based on the family of submodular inequalities is NPhard in general. For the wellknown subclass of flow cover inequalities, however, we show that if the client set is fixed, and if all capacities are equal, then the separation problem can be solved in polynomial time. For the flow cover inequalities based on an arbitrary client set, and for the effective capacity and single depot inequalities we develop separation heuristics. An important part of all these heuristic is based on constructive proofs that two specific conditions are necessary for the effective capacity inequalities to be facet defining. The proofs show precisely how structures that violate the two conditions can be modified to produce stronger inequalities. The family of combinatorial inequalities was originally developed for the uncapacitated facility location problem, but is also valid for the capacitated problem. No computational experience using the combinatorial inequalities has been reported so far. Here we suggest how partial output from the heuristic identifying violated submodular inequalities can be used as input to a heuristic identifying violated combinatorial inequalities. We report on computational results from solving 60 small and medium size problems.
On the TwoLevel Uncapacitated Facility Location Problem
 INFORMS J. COMPUT
, 1996
"... We study the twolevel uncapacitated facility location (TUFL) problem. Given two types of facilities, which we call yfacilities and zfacilities, the problem is to decide which facilities of both types to open, and to which pair of y and zfacilities each client should be assigned, in order to sat ..."
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Cited by 18 (3 self)
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We study the twolevel uncapacitated facility location (TUFL) problem. Given two types of facilities, which we call yfacilities and zfacilities, the problem is to decide which facilities of both types to open, and to which pair of y and zfacilities each client should be assigned, in order to satisfy the demand at maximum profit. We first present two multicommodity flow formulations of TUFL and investigate the relationship between these formulations and similar formulations of the onelevel uncapacitated facility location (UFL) problem. In particular, we show that all nontrivial facets for UFL define facets for the twolevel problem, and derive conditions when facets of TUFL are also facets for UFL. For both formulations of TUFL, we introduce new families of facets and valid inequalities and discuss the associated separation problems. We also characterize the extreme points of the LPrelaxation of the first formulation. While the LPrelaxation of a multicommodity formulation provi...
Valid inequalities and facets of the capacitated plant location problem
 Mathematical Programming
, 1989
"... Recently, several successful applications of strong cutting plane methods to combinatorial optimization problems have renewed interest in cutting plane methods, and polyhedral characterizations, of integer programming problems. In this paper, we investigate the polyhedral structure of the capacitate ..."
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Cited by 11 (1 self)
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Recently, several successful applications of strong cutting plane methods to combinatorial optimization problems have renewed interest in cutting plane methods, and polyhedral characterizations, of integer programming problems. In this paper, we investigate the polyhedral structure of the capacitated plant location problem. Our purpose is to identify facets and valid inequalities for a wide range of capacitated fixed charge problems that contain this prototype problem as a substructure. The first part of the paper introduces a family of facets for a version of the capacitated plant location problem with constant capacity K for all plants. These facet inequalities depend on K and thus differ fundamentally from the valid inequalities for the uncapacitated version of the problem. We also introduce a second formulation for a model with indivisible customer demand and show that it is equivalent to a vertex packing problem on a derived graph. We identify facets and valid inequalities for this version of the problem by applying known results for the vertex packing polytope.
A BranchandCut Algorithm for a Generalization of the Uncapacitated Facility Location Problem
 TOP
, 1995
"... We introduce a generalization of the wellknown Uncapacitated Facility Location Problem, in which clients can be served not only by single facilities but also by sets of facilities. The problem, called Generalized Uncapacitated Facility Location Problem (GUFLP), was inspired by the Index Selection P ..."
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Cited by 10 (2 self)
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We introduce a generalization of the wellknown Uncapacitated Facility Location Problem, in which clients can be served not only by single facilities but also by sets of facilities. The problem, called Generalized Uncapacitated Facility Location Problem (GUFLP), was inspired by the Index Selection Problem in physical database design. We formulate GUFLP as a Set Packing Problem, showing that our model contains all the clique inequalities (in polynomial number). Moreover, we describe an exact separation procedure for oddhole inequalities, based on the particular structure of the problem. These results are used within a branchandcut algorithm for the exact solution of GUFLP. Computational results on two different classes of test problems are given.
Adapting Polyhedral Properties from Facility to Hub Location Problems
 Discrete Applied Mathematics
, 2004
"... We examine the feasibility polyhedron of the uncapacitated hub location problem (UHL) with multiple allocation, which has applications in the fields of air passenger and cargo transportation, telecommunication and postal delivery services. In particular we determine the dimension and derive some cla ..."
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Cited by 7 (1 self)
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We examine the feasibility polyhedron of the uncapacitated hub location problem (UHL) with multiple allocation, which has applications in the fields of air passenger and cargo transportation, telecommunication and postal delivery services. In particular we determine the dimension and derive some classes of facets for this polyhedron. We develop a general rule about lifting facets from the uncapacitated facility location (UFL) problem to UHL. Using this lifting procedure we obtain a new class of facets for UHL which dominates the inequalities in the original formulation.
Polyhedral Techniques in Combinatorial Optimization II: Computations
 Statistica Neerlandica
, 1995
"... The polyhedral approach is one of the most powerful techniques available for solving hard combinatorial optimization problems. The main idea behind the technique is to consider the linear relaxation of the integer combinatorial optimization problem, and try to iteratively strengthen the linear formu ..."
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Cited by 6 (1 self)
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The polyhedral approach is one of the most powerful techniques available for solving hard combinatorial optimization problems. The main idea behind the technique is to consider the linear relaxation of the integer combinatorial optimization problem, and try to iteratively strengthen the linear formulation by adding violated strong valid inequalities, i.e., inequalities that are violated by the current fractional solution but satisfied by all feasible solutions, and that define highdimensional faces, preferably facets, of the convex hull of feasible solutions. If we have the complete description of the convex hull of feasible solutions all extreme points of this formulation are integral, which means that we can solve the problem as a linear programming problem. Linear programming problems are known to be computationally easy. In Part I of this article we discuss theoretical aspects of polyhedral techniques. Here we will mainly concentrate on the computational aspects. In particular we ...