3. Combinatorial Packing Problems

Abstract

3.1 Introduction This chapter investigates a certain class of combinatorial packing problems (CPPs) and some polyhedral relations between such problems and the set packing problem (SPP). Packing constraints are one of the most common problem characteristics in combinatorial optimization. They occur in path packing formulations of vehicle and crew scheduling problems, in Steiner tree packing approaches to VLSI and network design problems, and in coloring models of frequency assignment problems; see [38, 16] for surveys. The pure form of a packing problem is the SPP or stable set problem in a graph G = (V, E) with node weights w; it asks for a maximum weight set of mutually nonadjacent nodes. This problem has been studied extensively, and deep structural and algorithmic results have been achieved in areas such as antiblocking theory, the theory of perfect graphs, perfect and balanced matrix theory, and semidefinite programming (SDP); see [7, 20, 34, 8] for surveys. There is, in particular, a substantial structural and algorithmic knowledge of the set packing polytope, with many classes of strong and polynomial-time separable inequalities such as odd hole, odd antihole, and orthonormal representation constraints [35, 33, 44, 37, 20].