The Role of Integer Programming Techniques in Constraint Programming's Global Constraints

Abstract

Efforts aimed at combining operations research and constraint programming have become increasingly prominent and successful in the last few years. It is now widely recognized that integration, e.g., inference in the form of constraint propagation and relaxation in the form of linear programming, can yield substantial results. In this paper, we argue the benefits of constraint programming's global constraints as a basis for such an integration and discuss the advantages along with some examples. We illustrate the integration on the global cardinality structure, on piecewise linear functions, on variable subscripts, on the cycle structure and on resource constraints. Each example is completed with a case study.

A polyhedral approach to the alldifferent system

“…It follows that inequality (22) can be violated only if x j 1 + x j 3 < 2t + 1. In that case, inequality (22) could be violated…”

“…Such representations motivate the integration of CP and IP into a unified modeling and algorithmic framework offering the declarative simplicity of predicates coupled with a powerful engine based on mathematical and computational techniques (see [15,Chapter 1] for a more detailed discussion). Such a framework has been the main theme of several papers (see [4,13,14,16,22] and references contained there-in) while it is implemented in several programming environments (see, for example, OPL [28], Mosel [8], SCIP [1], etc.). The current work investigates the IP representation of the system of alldifferent predicates, thus contributing to the modeling and computational aptitude of such a framework.…”

Constraints and AI Planning

“…The IP and SAT approach will break the problems down into their specific frameworks and may then scan the resulting specifications for structures on which to apply specific solution strategies. But -although many efficient methods have been developed -the propositional clauses of SAT and the linear inequalities of IP are scarcely able to exploit the higher-level domain knowledge anymore to support search (see also [Milano00]). This is not the case for constraint programming, whose high-level constraints are able to capture domain-specific dependencies.…”

“…However, there are ongoing efforts to combine constraint programming with integer programming [Hooker00,Milano00,Refalo99].…”

SIMPL: A System for Integrating Optimization Techniques

“…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].…”