Searching for optimal integer solutions to set partitioning problems using column generation

Abstract

In this paper, we describe a new approach to increase the possibility of finding integer feasible columns to a set partitioning problem (SPP) directly in solving the linear programming (LP) relaxation using column generation. Traditionally, column generation is aimed to solve the LP-relaxation as quickly as possible without any concern for the integer properties of the columns formed. In our approach, we aim to generate columns forming an optimal integer solution while simultaneously solving the LP-relaxation. Using this approach, we can improve the possibility of finding integer solutions by heuristics at each node in the branch-and-bound search. In addition, we improve the possibility of finding high-quality integer solutions in cases where only the columns in the root node are used to solve the problem. The basis of our approach is a subgradient technique applied to a Lagrangian dual formulation of the SPP extended with an additional surrogate constraint. This extra constraint is not relaxed and is used to better control the subgradient evaluations and how the multiplier values are computed. The column generation is then directed, via the multipliers, to construct columns that form feasible integer solutions. Computational experiments show that we can generate optimal integer columns in a large set of well-known test problems as compared to both standard and stabilized column generation, and simultaneously keep the number of columns smaller than standard column generation. This is also supported by tests on a case study with work-shift generation.