Egon Balas

A lift-and-project cutting plane algorithm for mixed 0---1 programs

We propose a cutting plane algorithm for mixed 0–1 programs based on a family of polyhedra which strengthen the usual LP relaxation. We show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point. This cut is found through the solution of a linear program that has about twice the size of the usual LP relaxation. A lifting step is used to reduce the size of the LPs needed to generate the cuts. An additional strengthening step suggested by Balas and Jeroslow is then applied. We report our computational experience with a preliminary version of the algorithm. This approach is related to the work of Balas on disjunctive programming, the matrix cone relaxations of Lovász and Schrijver and the hierarchy of relaxations of Sherali and Adams.

Set Partitioning: A survey

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...

An Algorithm for Large Zero-One Knapsack Problems

We describe an algorithm for the 0-1 knapsack problem (KP), which relies mainly on three new ideas. The first one is to focus on what we call the core of the problem, namely, a knapsack problem equivalent to KP, defined on a particular subset of the variables. The size of this core is usually a small fraction of the full problem size, and does not seem to increase with the latter. While the core cannot be identified without solving KP, a satisfactory approximation can be found by solving the associated linear program (LKP). The second new ingredient is a binary search-type procedure for solving LKP which, unlike earlier methods, does not require any ordering of the variables. The computational effort involved in this procedure is linear in the number of variables. Finally, the third new feature is a simple heuristic which under certain conditions finds an optimal solution with a probability that increases with the size of KP. Computational experience with an algorithm based on the above ideas, on several ...

The prize collecting traveling salesman problem

The following is a valid model for an important class of scheduling and routing problems. A salesman who travels between pairs of cities at a cost depending only on the pair, gets a prize in every city that he vitis and pays a penalty to every city that he fails to visit, wishes to minimize his travel costs and net penalties, while visiting enough cities to collect a prescribed amount of prize money. We call this problem the Prize Collecting Traveling Salesman Problem (PCTSP). This paper discusses structural properties of the PCTS polytope, the convex hull of solutions to the PCTSP. In particular, it identifies several families of facet defining inequalities for this polytope. Some of these inequalities are related to facets of the ordinary TS polytope, others to facets of the knapsack polytope. They can be used in algorithms for the PCTSP either as cutting planes or as ingredients of a Lagrangean optimand.

Facets of the knapsack polytope

A necessary and sufficient condition is given for an inequality with coefficients 0 or 1 to define a facet of the knapsack polytope, i.e., of the convex hull of 0–1 points satisfying a given linear inequality. A sufficient condition is also established for a larger class of inequalities (with coefficients not restricted to 0 and 1) to define a facet for the same polytope, and a procedure is given for generating all facets in the above two classes. The procedure can be viewed as a way of generating cutting planes for 0–1 programs.

Disjunctive Programming and a Hierarchy of Relaxations for Discrete Optimization Problems

We discuss a new conceptual framework for the convexification of discrete optimization problems, and a general technique for obtaining approximations to the convex hull of the feasible set. The concepts come from disjunctive programming and the key tool is a description of the convex hull of a union of polyhedra in terms of a higher dimensional polyhedron. Although this description was known for several years, only recently was it shown by Jeroslow and Lowe to yield improved representations of discrete optimization problems. We express the feasible set of a discrete optimization problem as the intersection (conjunction) of unions of polyhedra, and define an operation that takes one such expression into another, equivalent one, with fewer conjuncts. We then introduce a class of relaxations based on replacing each conjunct (union of polyhedra) by its convex hull. The strength of the relaxations increases as the number of conjuncts decreases, and the class of relaxations forms a hierarchy that spans the spec...

Disjunctive programming: properties of the convex hull of feasible points

In this paper we characterize the convex hull of feasible points for a disjunctive program, a class of problems which subsumes pure and mixed integer programs and many other nonconvex programming problems. Two representations are given for the convex hull of feasible points, each of which provides linear programming equivalents of the disjunctive program. The first one involves a number of new variables proportional to the number of terms in the disjunctive normal form of the logical constraints; the second one involves only the original variables and the facets of the convex hull. Among other results, we give necessary and sufficient conditions for an inequality to define a facet of the convex hull of feasible points. For the class of disjunctive programs that we call facial, we establish a property which makes it possible to obtain the convex hull of points satisfying n disjunctions, in a sequence of n steps, where each step generates the convex hull of points satisfying one disjunction only.

Finding a maximum clique in an arbitrary graph

We describe a new type of branch and bound procedure for finding a maximum clique in an arbitrary graph

Pivot and Complement–A Heuristic for 0-1 Programming

G = (V,E)

INTERSECTION CUTS-A NEW TYPE OF CUTTING PLANES FOR INTEGER PROGRAMMING

. The two main ingredients, both of