Manfred W. Padberg

A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems

An algorithm is described for solving large-scale instances of the Symmetric Traveling Salesman Problem (STSP) to optimality. The core of the algorithm is a “polyhedral” cutting-plane procedure that exploits a subset of the system of linear inequalities defining the convex hull of the incidence vectors of the hamiltonian cycles of a complete graph. The cuts are generated by several identification procedures that have been described in a companion paper. Whenever the cutting-plane procedure does not terminate with an optimal solution the algorithm uses a tree-search strategy that, as opposed to branch-and-bound, keeps on producing cuts after branching. The algorithm has been implemented in FORTRAN. Two different linear programming (LP) packages have been used as the LP solver. The implementation of the algorithm and the interface with one of the LP solvers is described in sufficient detail to permit the replication of our experiments. Computational results are reported with up to 42 STSPs with sizes rangin...

Solving Large-Scale Zero-One Linear Programming Problems

In this paper we report on the solution to optimality of 10 large-scale zero-one linear programming problems. All problem data come from real-world industrial applications and are characterized by sparse constraint matrices with rational data. About half of the sample problems have no apparent special structure; the remainder show structural characteristics that our computational procedures do not exploit directly. By todays standards, our methodology produced impressive computational results, particularly on sparse problems having no apparent special structure. The computational results on problems with up to 2,750 variables strongly confirm our hypothesis that a combination of problem preprocessing, cutting planes, and clever branch-and-bound techniques permit the optimization of sparse large-scale zero-one linear programming problems, even those with no apparent special structure, in reasonable computation times. Our results indicate that cutting-planes related to the facets of the underlying polytope are an indispensable tool for the exact solution of this class of problem. To arrive at these conclusions, we designed an experimental computer system PIPX that uses the IBM linear programming system MPSX/370 and the IBM integer programming system MIP/370 as building blocks. The entire system is automatic and requires no manual intervention.

On the facial structure of set packing polyhedra

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 “⩽”. 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.

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...

Odd Minimum Cut-Sets and b-Matchings

We show that the determination of a minimum cut-set of odd cardinality in a graph with even and odd vertices can be dealt with by a minor modification of the polynomially bounded algorithm of Gomory and Hu for multi-terminal networks. We connect this problem to the problem of identifying a matching or blossom constraint that chops off a point which is not contained in the convex hull of matchings or proving that no such inequality exists. Both the b-matching problems without and with upper bounds are considered. We discuss how the results of this paper can be used in conjunction with commercial LP packages lo solve b-matching problems.

The Boolean quadric polytope: some characteristics, facets and relatives

We study unconstrained quadratic zero–one programming problems havingn variables from a polyhedral point of view by considering the Boolean quadric polytope QPn inn(n+1)/2 dimensions that results from the linearization of the quadratic form. We show that QPn has a diameter of one, descriptively identify three families of facets of QPn and show that QPn is symmetric in the sense that all facets of QPn can be obtained from those that contain the origin by way of a mapping. The naive linear programming relaxation QPnLP of QPn is shown to possess the Trubin-property and its extreme points are shown to be {0,1/2,1}-valued. Furthermore, O(n3) facet-defining inequalities of QPn suffice to cut off all fractional vertices of QPnLP, whereas the family of facets described by us has at least O(3n) members. The problem is also studied for sparse quadratic forms (i.e. when several or many coefficients are zero) and conditions are given for the previous results to carry over to this case. Polynomially solvable problem instances are discussed and it is shown that the known polynomiality result for the maximization of nonnegative quadratic forms can be obtained by simply rounding the solution to the linear programming relaxation. In the case that the graph induced by the nonzero coefficients of the quadratic form is series-parallel, a complete linear description of the associated Boolean quadric polytope is given. The relationship of the Boolean quadric polytope associated to sparse quadratic forms with the vertex-packing polytope is discussed as well.

Perfect zero---one matrices

A zero–one matrix is called perfect if the polytope of the associated set packing problem has integral vertices only. By this definition, all totally unimodular zero–one matrices are perfect. In this paper we give a characterization of perfect zero–one matrices in terms offorbidden submatrices. Perfect zero–one matrices are closely related to perfect graphs and constitute a generalization of balanced matrices as introduced by C. Berge. Furthermore, the results obtained here bear on an unsolved problem in graph theory, the strong perfect graph conjecture, also due to C. Berge.

On the symmetric travelling salesman problem I: Inequalities

We investigate several classes of inequalities for the symmetric travelling salesman problem with respect to their facet-defining properties for the associated polytope. A new class of inequalities called comb inequalities is derived and their number shown to grow much faster with the number of cities than the exponentially growing number of subtour-elimination constraints. The dimension of the travelling salesman polytope is calculated and several inequalities are shown to define facets of the polytope. In part II (“On the travelling salesman problem II: Lifting theorems and facets”) we prove that all subtour-elimination and all comb inequalities define facets of the symmetric travelling salesman polytope.

On the Set-Covering Problem

This paper establishes some useful properties of the equality-constrained set-covering problem P and the associated linear program P′. First, the Dantzig property of transportation matrices is shown to hold for a more general class of matrices arising in connection with adjacent integer solutions to P′. Next, we show that, for every feasible integer basis to P′, there are at least as many adjacent feasible integer bases as there are nonbasic columns. Finally, given any two basic feasible integer solutions x1 and x2 to P′, x2 can be obtained from x1 by a sequence of p pivots (where p is the number of indices j ϵ N for which xj1 is nonbasic and xj2 = 1), such that each solution in the associated sequence is feasible and integer. Some of our results have been conjectured earlier by Andrew, Hoffmann, and Krabek in a paper presented to ORSA in 1968.

Approximating separable nonlinear functions via mixed zero-one programs

We discuss two models from the literature that have been developed to formulate piecewise linear approximation of separable nonlinear functions by way of mixed-integer programs. We show that the most commonly proposed method is computationally inferior to a lesser known technique by comparing analytically the linear programming relaxations of the two formulations. A third way of formulating the problem, that shares the advantages of the better of the two known methods, is also proposed.