Heuristic method for the set covering problem

We present a Lagrangian-based heuristic for the well-known Set Covering Problem (SCP). The algorithm was initially designed for solving very large scale SCP instances, involving up to 5,000 rows and 1,000,000 columns, arising from crew scheduling in the Italian Railway Company, Ferrovie dello Stato SpA. In 1994 Ferrovie dello Stato SpA, jointly with the Italian Operational Research Society, organized a competition, called FASTER, intended to promote the development of algorithms capable of producing good solutions for these instances, since the classical approaches meet with considerable difficulties in tackling them. The main characteristics of the algorithm we propose are (1) a dynamic pricing scheme for the variables, akin to that used for solving large-scale LPs, to be coupled with subgradient optimization and greedy algorithms, and (2) the systematic use of column fixing to obtain improved solutions. Moreover, we propose a number of improvements on the standard way of defining the step-size and the ascent direction within the subgradient optimization procedure, and the scores within the greedy algorithms. Finally, an effective refining procedure is proposed. Our code won the first prize in the FASTER competition, giving the best solution value for all the proposed instances. The algorithm was also tested on the test instances from the literature: in 92 out of the 94 instances in our test bed we found, within short computing time, the optimal (or the best known) solution. Moreover, among 18 instances for which the optimum is not known, in 6 cases our solution is better than any other solution found by previous techniques.