A decomposition method for valued CSPs

2009

Harmonization with four voices is a musical problem which is subject to hard constraints, which absolutely need to be fulfilled, as well as to soft constraints, which preferably hold, but are not mandatory. In this paper, we model this problem as a valued constraint satisfaction problem (VCSP): costs are assigned to possible solutions based on the constraints they violate. We design an algorithm that finds a minimal-cost solution, thus solving the harmonization problem, and we present initial results obtained by this algorithm.

The objective of the Maximal Constraint Satisfaction Problem (Max-CSP) is to find an instantiation which minimizes the number of constraint violations in a constraint network. In this paper, inspired from the concept of inferred disjunctive constraints introduced by Freuder and Hubbe, we show that it is possible to exploit the arc-inconsistency counts, associated with each value of a network, in order to avoid exploring useless portions of the search space. The principle is to reason from the distance between the two best values in the domain of a variable, according to such counts. From this reasoning, we can build a decomposition technique which can be used throughout search in order to decompose the current problem into easier sub-problems. Interestingly, this approach does not depend on the structure of the constraint graph, as it is usually proposed. Alternatively, we can dynamically post hard constraints that can be used locally to prune the search space. The practical interest of our approach is illustrated, using this alternative, with an experimentation based on a classical branch and bound algorithm, namely PFC-MRDAC.

2004

Nowadays, many real problems can be modeled as Constraint Satisfaction Problems (CSPs). Generally, these problems are solved by search algorithms, which require an order in which variables and values should be considered. Choosing the right order of variables and values can noticeably improve the efficiency of constraint satisfaction. The order in which constraints are studied can also improve efficiency, particularly in problems with non-binary constraints.

2008

This article introduces constraint integer programming (CIP), which is a novel way to combine constraint programming (CP) and mixed integer programming (MIP) methodologies. CIP is a generalization of MIP that supports the notion of general constraints as in CP. This ap- proach is supported by the CIP framework SCIP, which also integrates techniques for solving satisability problems. SCIP is available

A constraint satisfaction problem, or CSP, can be reformulated as an integer linear programming problem. The reformulated problem can be solved via polynomial multiplication. If the CSP has n variables whose domain size is m, and if the equivalent programming problem involves M equations, then the number of solutions can be determined in time O(nm2 M −n ). This surprising link between search problems and algebraic techniques allows us to show improved bounds for several constraint satisfaction problems, including new simply exponential bounds for determining the number of solutions to the n-queens problem. We also address the problem of minimizing M for a particular CSP.

Lecture Notes in Computer Science, 2010

We report on experiments with turning the branch-price-andcut framework SCIP into a generic branch-price-and-cut solver. That is, given a mixed integer program (MIP), our code performs a Dantzig-Wolfe decomposition according to the user's specification, and solves the resulting re-formulation via branch-and-price. We take care of the column generation subproblems which are solved as MIPs themselves, branch and cut on the original variables (when this is appropriate), aggregate identical subproblems, etc. The charm of building on a well-maintained framework lies in avoiding to re-implement state-of-the-art MIP solving features like pseudo-cost branching, preprocessing, domain propagation, primal heuristics, cutting plane separation etc.

Discrete Applied Mathematics, 1980

A concept of equi-assignment is introduced and characterized by means of a property based on the Euclidean algorithm fat findir 2 l..e greatest common divisor. Then a set of optimal solutions of a particu!ar discrete optimization problem is found by determining a suitable equi-assignment. Throughout this paper Z+ and lE3 will denote the set of all nonnegative integers and the set (0, l}, respectively. Define @= fill& where lEI"=BNE3~ l 0 l NB. n=-*The results of this pap er were presented at the Tenth Symposium or1 Mathematical Fkogramming (MONTREAL-1979).

Constraints, 2002

Abstract. We consider the Weighted Constraint Satisfaction Problem which is an important problem in Artificial Intelligence. Given a set of variables, their domains and a set of constraints between variables, our goal is to obtain an assignment of the variables to domain values such that ...

2007

Constraint Programming is a powerful approach for modeling and solving many combinatorial problems, scalability, however, remains an issue in practice. Abstraction and reformulation techniques are often sought to overcome the complexity barrier. In this paper we introduce four reformulation techniques that operate on the various components of a Constraint Satisfaction Problem (CSP) in order to reduce the cost of problem solving and facilitate scalability.

2006

Abstract Set variables are ubiquitous in modeling (soft) constraint problems, but efforts on practical consistency algorithms for Weighted Constraint Satisfaction Problems (WCSPs) have only been on integer variables. We adapt the classical notion of set bounds consistency for WC-SPs, and propose efficient representation schemes for set variables and common unary, binary, and ternary set constraints, as well as cardinality constraints.