A New Formulation of the Set Covering Problem for Metaheuristic Approaches

Abstract

Two difficulties arise when solving the set covering problem (SCP) with metaheuristic approaches: solution infeasibility and set redundancy. In this paper, we first present a review and analysis of the heuristic approaches that have been used in the literature to address these difficulties. We then present a new formulation that can be used to solve the SCP as an unconstrained optimization problem and that eliminates the need to address the infeasibility and set redundancy issues. We show that all local optimums with respect to the new formulation and a 1-flip neighbourhood structure are feasible and free of redundant sets. In addition, we adapt an existing greedy heuristic for the SCP to the new formulation and compare the adapted heuristic to the original heuristic using 88 known test problems for the SCP. Computational results show that the adapted heuristic finds better results than the original heuristic on most of the test problems in shorter computation times.

A Heuristic Approach for Minimum Set Cover Problem

“…The 3rd and 4th column represent the solution found using greedy and LP rounding approach. The 5th and 6th column represent the solutions found in [5] and [7]. The last two columns contain the result found using proposed approach, started from greedy approach and LP rounding approach respectively.…”

“…Summary: The optimal solution presented in Table II and Table III are taken from [7]. The quality of a solution derived by an algorithm is measured by Quality Ratio which is defined as a ratio of the derived solution to the optimal solution.…”

“…The optimization has numerous applications in different areas of studies and industrial applications [3]. The applications include multiple sequence alignments for computational biochemistry, manufacturing, network security, service planning and location problems [4]- [7].…”

A Generic Algorithmic Framework to Solve Special Versions of the Set Partitioning Problem

“…The SPP is usually tackled by one of the following strategies or by mixing several of them in a portfolio approach (for more detailed surveys regarding optimization algorithms for the SPP over the last forty years, see [6,8,10,14,24,25]). …”

“…Heuristics limiting the search space in some way have also been proposed to find suboptimal solutions in reasonable time, including genetic algorithms [14], dual ascent [10], simulated annealing, and neural networks [24], and other metaheuristic algorithms [8]. However, such approaches do not provide any worstcase guarantee regarding the closeness to optimality [45].…”