Set Partitioning: A survey

Abstract

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 integer, and we describe the facial structure of this polytope to the extent that it is known. In this description, the one-to-one correspondence between graphs and set packing polytopes plays a central role the facets of a set packing polytope are related to certain subgraphs of an associated graph. We review the various classes of facets and associated subgraphs that have been identified to date. Since facets of the set packing polytope can be arbitrarily complex and therefore computationally expensive to generate, we then discuss a class of inequalities derived from the disjunctive conditions of the set partitioning problem, which are facets of various relaxations of the set partitioning polytope, and can easily be computed from a fractional simplex tableau. These inequalities can be used in simplex-based fractional cutting plane algorithms. Another class of inequalities, also derived from the logical implications of the set partitioning constraints, is unrelated to any particular simplex tableau. These inequalities have coefficients of 0, 1 or -1 and provide convenient all integer cutting planes especially for a primal approach. Finally, we characterize adjacency relations between vertices of the set partitioning and set packing polytopes, on these polytopes as well as on their linear programming relaxations. The basic property that every edge of the set partitioning (set packing) polytope is also an edge of the linear programming relaxation of the latter, is viewed in the context of the need for appropriate criteria to identify such edges that meet a given vertex. All this theory is relatively new, a product of the last five years. Proofs are in general omitted, but sources are referenced in each case. Part 3 focuses on algorithms. We first discuss the two main types that are by now well established, implicit enumeration and (traditional) cutting planes. While in the first category several specialized algorithms have been developed, of which we discuss the ones that to our knowledge have been tested and found successful, algorithms in the second category are basically nonspecialized. Nevertheless, since cutting planes are known to be relatively efficient on set partitioning problems, we discuss some features of the algorithms and codes in this class. For both of these approaches, we briefly review the published computational experience. Next we discuss some recently developed approaches which are either untested or tested to a very limited extent, but which are based on new ideas that seem to hold some promise. The first one is a column generating procedure, based on the adjacency properties discussed in Part 2. It uses a modified all-integer version of the primal simplex algorithm, and generates composite columns corresponding to edges of the feasible set which connect a given integer vertex to a better one. The second procedure is a hybrid algorithm which combines a primal cutting plane method based on all integer cuts with coefficients 0, 1 or -1, with implicit enumeration applied to subproblems so defined as to generate an improvement at each iteration. The third approach uses a new symmetric subgradient method to solve the set partitioning linear program, amended with cutting planes, in an attempt to eliminate the difficulties involved in solving these large, very constrained and very degenerate linear programs by the simplex method. Finally, the fourth one deals with the set partitioning problem via an equivalent weighted node covering problem, which it solves by a hybrid cutting plane-branch and bound algorithm. The latter again avoids recourse to the simplex method, and uses a labeling technique instead. We assume some familiarity on the part of the reader with the basic concepts of graph theory. For background material in this field, the reader is referred to Harary (1969), Berge (1970), Roy (1969), (1970), Christofides (1975). For background in the general areas of linear and integer programming, see Dantzig (1963), Simmonard (1966) for the former and Garfinkel and Nemhauser (1972) for the latter.