Cooperation of LP Solvers for Solving MILPs

A standard approach to solve Mixed Integer Linear Programs is to perform a global branch and bound search through all possible combinations. Due to the hardness of the problem, this search must be closely controlled by a constraint solver which uses constraints to prune the search space in an a priori way. In this paper, we deene a new domain reduction solver which uses in a cooperative way a set of Linear Programming solvers. The idea is to compute the actual range of values of the integer variables with respect to the continuous relaxation of the problem, and then narrow these domains to the closest integer interval. This narrowing is iteratively performed until a xed point is reached where all domains are bound by integer values which belong to the continuous relaxation of the problem. This xed point corresponds to a new partial consistency, which is stronger than the continuous relaxation and allows us to solve MILPs more eeciently. 1 Motivations MILPs (Mixed Integer Linear Programs) have many industrial applications such as network design, crew scheduling, resource allocation or cutting stock problems. The general form of a MILP is displayed in gure 1. minimize P n j=1 c j x j subject to P n j=1 a ij x j b i 8i 2 1::m l j x j u j 8j 2 1::n x j integral 8x j 2 X int Figure 1: General form of a MILP In this MILP, X int is the subset of variables that are constrained to have integer values. When dropping these integrality restrictions, the resulting linear program |called the continuous relaxation| can easily be solved (by using the simplex algorithm for instance). However, when integrality constraints must be taken into account, the problem becomes much harder (NP-complete). A standard approach to solve MILPs is to perform a global branch and bound search 14] (also called branch and relax search in 11]). The idea is to solve a relaxation of the MILP and then, if some of the relaxed constraints are not satissed, to split the MILP into sub-MILPs on which this process is repeated. The relaxation step is performed at each node of the search in order to detect infeasibility and prune the search space in an a priori way. Most of the time, this relaxation step is performed by a Linear Programming solver (LP solver) which veriies that the continuous relaxation …