Showing papers on "Set cover problem published in 1980"

Set covering algorithms using cutting planes, heuristics, and subgradient optimization: A computational study

Abstract: We report on the implementation and computational testing of several versions of a set covering algorithm, based on the family of cutting planes from conditional bounds discussed in the companion paper [2] The algorithm uses a set of heuristics to find prime covers, another set of heuristics to find feasible solutions to the dual linear program which are needed to generate cuts, and subgradient optimization to find lower bounds It also uses implicit enumeration with some new branching rules Each of the ingredients was implemented and tested in several versions The variant of the algorithm that emerged as best was run on 55 randomly generated test problems (20 of them from the literature), with up to 200 constraints and 2000 variables The results show the algorithm to be more reliable and efficient than earlier procedures on large, sparse set covering problems

Cutting planes from conditional bounds: A new approach to set covering

Abstract: A conditional lower bound on the minimand of an integer program is a number which would be a valid lower bound if the constraint set were amended by certain inequalities, also called conditional If such a conditional lower bound exceeds some known upper bound, then every solution better than the one corresponding to the upper bound violates at least one of the conditional inequalities This yields a valid disjunction, which can be used to partition the feasible set, or to derive a family of valid cutting planes In the case of a set covering problem, these cutting planes are themselves of the set covering type The family of valid inequalities derived from conditional bounds subsumes as a special case the Bellmore-Ratliff inequalities generated via involutory bases, but is richer than the latter class and contains considerably stronger members, where strength is measured by the number of positive coefficients We discuss the properties of the family of cuts from conditional bounds, and give a procedure for generating strong members of the family Finally, we outline a class of algorithms based on these cuts Our approach was implemented and extensively tested in a computational study whose results are reported in a companion paper [2] The algorithm that emerged from the testing seems capable of solving considerably larger set covering problem than earlier methods

A Note On Some Computationally Difficult Set Covering Problems

Abstract: Fulkerson et al. have given two examples of set covering problems that are empirically difficult to solve. They arise from Steiner triple systems and the larger problem, which has a constraint matrix of size 330 × 45 has only recently been solved. In this note, we show that the Steiner triple systems do indeed give rise to a series of problems that are probably hard to solve by implicit enumeration. The main result is that for ann variable problem, branch and bound algorithms using a linear programming relaxation, and/or elimination by dominance require the examination of a super-polynomial number of partial solutions