Hybrid heuristic algorithms for Set Covering

Abstract

The paper considers the minimal set covering problem. It is one of the most important discrete optimization problems because it serves as model for real-world problems, such as the best utilization of resources and workers in several fields. In general terms, there are a lot of tasks to do and a lot of resources for doing these tasks. One should find the subset of resources of minimal cost that fulfils all tasks. Of course, since the problem pertains to discrete optimization, the well-known operations research tool of integer programming is a natural choice. However, the problem is very hard to solve even by current supercomputers since the computing time grows exponentially with the problem size when a naive representation in the mathematical field of integer programming is adopted. Thus, a novel representation is described which combines two approaches for achieving good computation performance in large-size problems: first a linear programming model, then an improvement technique based on a simulated neural network. The linear programming relaxation model is new in that it uses a constraint matrix which is not problem instance specific as in the traditional model, and all problem specific data are incorporated in the objective function vector. Only two types of general constraints are used: choice constraints, which ensure each ‘job’ is covered, and exclusion constraints, which prevent redundant resources to be used. Significant test results and comparisons to other algorithms are reported. They show that the proposed new algorithm is effective in solving practical cases. In particular, it reports an average improvement in the execution time over the current best algorithms in solving hard problems while producing the same order of solution quality, i.e. optimality approximation. Minimal set covering (MSC) is a known NP-hard problem. It is the model of many important real-world problems and plays a central role in rostering (crew scheduling) problems. Large-size set covering problems (constraint matrix of about 10,000∗100,000 elements) are considered.First, the traditional model of MSC as integer program is recalled, then a new representation in linear programming (LP) is described. Finally, a new algorithm based on recurrent neural networks is introduced for the optimization of the pivot positions selection. This algorithm is combined with the LP algorithm to guarantee an optimal solution. The heuristic part serves for choosing the pivot positions in the Simplex algorithm for LP. The results of significant tests on real data are reported. They compare favourably with the best-known results on set covering, both in terms of execution time and solution accuracy.