An efficient heuristic for large set covering problems

Abstract

A heuristic solution procedure for set covering is presented that works well for large, relatively dense problems. In addition, a confidence interval is established about the unknown global optimum. Results are presented for 30 large randomly generated problems. INTRODUCTIONThe set covering problem is a combinatorial optimization problem that was shown by Karp [9] to be NP-complete. Since there are many applications of set covering in scheduling, assembly-line balancing, capital investment, switching theory, and information retrieval that involve the solution of large set covering problems, there exists the need for heuristic procedures capable of efficiently obtaining near-optimal solutions to these large problems.In this article we develop a procedure that quickly generates a good solution to large set covering problems, and then we use the interval estimation procedure developed by Golden and Alt [7] on the traveling salesman problem to construct a confidence interval about the unknown global optimum. Finally, we test these procedures on 30 large randomly generated problems. BACKGROUND

A lagrangian heuristic for set-covering problems

“…All of the test problems (except problem set VW) have been produced using the scheme of Balas and Ho [3], namely, column costs (c,) are integers randomly generated from [l, 1001; every column covers at least one row; every row is covered by at least two columns. In order to compare the heuristic algorithm presented in this article with other heuristics we also coded in FORTRAN the BH heuristic of Balas and Ho [3] involving using five different functions f(j, c,, S ) and choosing the best solution found; the SCHEURI heuristic of Vasko and Wilson [17] involving using seven different functions f(j, c,, S ) and choosing the best solution found; and the SCFUNClT07 heuristic of Vasko and Wilson [17] involving a random choice from seven different functions f(j, c,, S ) (we executed SCFUNClT07 ten times and choose the best solution found).…”

“…Note here that we modified the Vasko and Wilson [17] heuristics (SCHEURI and SCFUNClT07) slightly in that the final step in both of these heuristics involves a 1-opt neighborhood search on a feasible solution S to the original SCP. Given the usual definition of a 1-opt neighborhood search this means that the number of columns in the solution (ISl) cannot change.…”

An efficient local search heuristic with row weighting for the unicost set covering problem

“…The simplest one for SCPs is the greedy algorithm (Chvatal, 1979). Later, several randomized greedy algorithms (Feo & Resende, 1989;Vasko, 1984) are proposed. They usually produce better results than the deterministic greedy one.…”

Big Step Greedy Heuristic for Maximum Coverage Problem

“…Random and probabilistic greedy approximate algorithms [7,8,9] produce better solutions than the classical greedy algorithm for set covering problem. Randomized greedy algorithm used by Grossman and Wool [2] is same as classical greedy algorithm except that ties are broken at random and the basic algorithm is repeated N times and returns the best solution among the N solutions.…”