A 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 diiculties 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 LP's, to be coupled with subgradient optimization and greedy algorithms, and (2) the systematic use of column xing to obtain improved solutions. Moreover, we propose a number of improvements on the standard way of deening the step-size and the ascent direction within the subgradient optimization procedure, and the scores within the greedy algorithms. Finally, an eeective reening procedure is proposed. Our code won the rst 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 the 18 instances for which the optimum is not known, in 6 cases our solution is better than any other solution found by previous techniques. The Set Covering Problem (SCP) is a main model for several important applications, including crew scheduling, where a given set of trips has to be covered by a minimum-cost set of pairings, a pairing being a sequence of trips that can be performed by a single crew. A widely-used approach to crew scheduling works as follows. First, a very large number of pairings are generated. Then an SCP is solved, having as row set the trips to be covered, and as column set the pairings generated. In railway applications, very large scale SCP instances typically arise, involving thousands of rows and millions of columns. The classical methods proposed for SCP meet with considerable diiculties in tackling these 1