Showing papers on "Set cover problem published in 1977"

The Set-Covering Problem: A New Implicit Enumeration Algorithm

Abstract: The set-covering problem is to minimize cx subject to Ax ≧ en over all binary n-vectors x. A is an m × n binary matrix and en is an n-vector of 1's. We develop a new implicit enumeration strategy to solve this problem. The branching strategy is similar to the row partitioning strategy used by other authors in the partitioning problem. Simple and sharp bounds are obtained by relaxing the constraints of the associated linear program by attaching nonnegative multipliers to them. Good multipliers are obtained by using the subgradient optimization technique. Computational experience shows that these bounds are at least one order of magnitude more efficient than the ones obtained by solving the associated linear program with the simplex method. Computational results with this new implicit enumeration algorithm are encouraging. Problems with as many as 50 constraints and 100 variables were solved in the order of 100 seconds of CPU time on an IBM 360-67.

The Solution of the Graph-Coloring Problem as a Set-Covering Problem

Abstract: The minimum number of channels required to satisfy the frequency assignment constraints of a collection of equipments for which specified pairs of equipments are denied cochannel operation has been shown to be equal to the chromatic number of an associated graph. This paper provides a formulation of the graph-coloring problem in terms of a related problem for which efficient computer programs have been prepared. This related problem is known as the zero-one set covering problem in which the rows of a binary array are "covered" with the minimum number of columns such that every row contains at least one nonzero element in a column belonging to the cover.

The Sequential Covering Problem Under Uncertainty

Abstract: The formulation of the set covering problem in the literature is under the assumption that its binary variables are deterministic. In this paper we consider a randomized set covering problem where there is a positive probability that a decision to include a variable in the cover (xi = 1) will in fact be realized, because of failure, as ineffective (xi = 0). Furthermore, in the present formulation of the covering problems, the values of all variables are determined simultaneously. We allow a sequential selection of variables and present an efficient heuristic procedure which minimizes the expected costs necessary for the construction of a cover. An example for such problem is the detection of the state of a system by a sequential testing of its interrelated elements. Each element may have different testing costs and its functioning probability is independent of others. More applications of this model are described in the paper, and computational experience is reported.