This paper discusses the set partitioning or equality-constrained set covering problem. It is a survey of theoretical results and solution methods for this problem, and while we have tried not to omit anything important, we have no claim to completeness. Critical comments pointing out possible omissions or misstatements will be welcome.Part 1 gives some background material. It starts by discussing the uses of the set partitioning model; then it introduces the concepts to be used throughout the paper, and connects our problem to its close and distant relatives which play or may play a role in dealing with it: set packing and set covering, edge matching and edge covering, node packing and node covering, clique covering. The crucial equivalence between set packing/partitioning and node packing problems is introduced.Part 2 deals with structural properties of the set packing and set partitioning polytopes. We discuss necessary and sufficient conditions for all vertices of the set packing polytope to be intege...

Set Partitioning: A survey

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

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

We present an algorithm for a mixed set covering/partitioning model that includes as special cases the well-known set covering problem and set partitioning problem. The novel feature of our algorit...

https://www.jstor.org/stable/pdfplus/2631900.pdf

Optimal solution of set covering/partitioning problems using dual heuristics

Pivot and Complement is a heuristic for finding approximate solutions to 0-1 programming problems. It uses the fact that a 0-1 program is equivalent to the associated linear program with the added requirement that all slack variables, other than those in the upper bounding constraints, be basic. The procedure starts by solving the linear program; then performs a sequence of pivots aimed at putting all slacks into the basis at a minimal cost; finally, it attempts to improve the 0-1 solution obtained in this way by a local search based on complementing certain sets of 0-1 variables. The computational effort involved in the procedure is bounded by a polynomial in the number of constraints and variables. For the 92 test problems with 5 to 200 constraints and 20 to 900 variables on which the procedure was run, the time used to solve the linear program on the average considerably exceeded the time used for everything else. As to the quality of the solutions obtained, in 45 cases, or 49% of the total, the proced...

https://www.jstor.org/stable/info/2630747

Pivot and Complement–A Heuristic for 0-1 Programming

Enhancing an algorithm for set covering problems

Computational experience with a modified linear programming method for the inequality or equality set covering problem (i.e. minimize cx subject to Ex ≥ e or Ex = e, xi = 0 or 1, where E is a zero-one matrix, e is a column of ones, and c is a nonnegative integral row) is presented. The zero-one composition of the constraint matrix and the right-hand side of ones suggested an algorithm in which dual simplex iterations are performed whenever unit pivots are available and Gomory all integer cuts are adjoined when they are not. Applications to enumerative and heuristic schemes are also discussed.

Ronald D. Koncal

Papers

Set Covering by an All Integer Algorithm: Computational Experience