In this paper we address ourselves to identifying facets of the set packing polyhedron, i.e., of the convex hull of integer solutions to the set covering problem with equality constraints and/or constraints of the form "ź". This is done by using the equivalent node-packing problem derived from the intersection graph associated with the problem under consideration. First, we show that the cliques of the intersection graph provide a first set of facets for the polyhedron in question. Second, it is shown that the cycles without chords of odd length of the intersection graph give rise to a further set of facets. A rather strong geometric property of this set of facets is exhibited.

On the facial structure of set packing polyhedra

We consider two convex polyhedra related to the vertex packing problem for a finite, undirected, loopless graphG with no multiple edges. A characterization is given for the extreme points of the polyhedron\(\mathcal{L}_G = \{ x \in R^n :Ax \leqslant 1_m ,x \geqslant 0\} \), whereA is them × n edge-vertex incidence matrix ofG and 1m is anm-vector of ones. A general class of facets of

= convex hull{x∈Rn:Ax≤1m,x binary} is described which subsumes a class examined by Padberg [13]. Some of the results for

are extended to a more general class of integer polyhedra obtained from independence systems.

Properties of vertex packing and independence system polyhedra

This paper presents a new approach to the solution of multi-target tracking problems. 0-1 integer programming methods are used to alleviate the combinatorial computing difficulties that accompany any but the smallest of such problems. Multitarget tracking is approached as an unsupervised pattern recognition problem. A multiple-hypothesis test is performed to determine which particular combination of the many feasible tracks is most likely to represent actual targets. This multiple hypothesis test is shown to have the computational structure of the set packing and set partitioning problems of 0-1 integer programming. Multitarget tracking problems that are translated into this form can be rapidly solved, using well-known discrete optimization techniques such as implicit enumeration.

Application of 0-1 integer programming to multitarget tracking problems

We survey some of the recent results that have been obtained in connection with covering, packing, and knapsack problems formulated as linear programming problems in zero-one variables.

Covering, Packing and Knapsack Problems

The uncapacitated plant location problem is considered as a node-packing problem. For this problem, several valid inequalities and facets are discussed. Necessary and sufficient conditions for trivial facets along with necessary conditions for nontrivial facets are derived. In addition, all of the facets for the case of three plants and three or more destinations are identified.

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On the Uncapacitated Plant Location Problem. I: Valid Inequalities and Facets

Publisher Summary This chapter discusses integer programming by emphasizing computation and relations among models. Integer programming deals with the class of mathematical programming problems in which some or all of the variables are required to be integers. The chapter discusses the case in which both the objective function and constraints are linear. It describes integer linear program (ILP) models and some relationships among integer programming models and presents the classification of them according to computational complexity. A difficulty in reporting computational results is the number of different computers used. There is a problem in reporting results on approximate methods. There is virtually no point in reporting times to find a suboptimal solution without also giving its deviation from optimality. Unfortunately, for the large problems for which approximate methods are designed, optimal solutions are generally unknown. The chapter describes the most widely-used algorithms and presents a general enumeration algorithm for the mathematical-programming problem.

R.S. Garfinkel

Papers

A Survey of Integer Programming Emphasizing Computation and Relations among Models