Leslie E. Trotter

On the capacitated vehicle routing problem

Abstract. We consider the Vehicle Routing Problem, in which a fixed fleet of delivery vehicles of uniform capacity must service known customer demands for a single commodity from a common depot at minimum transit cost. This difficult combinatorial problem contains both the Bin Packing Problem and the Traveling Salesman Problem (TSP) as special cases and conceptually lies at the intersection of these two well-studied problems. The capacity constraints of the integer programming formulation of this routing model provide the link between the underlying routing and packing structures. We describe a decomposition-based separation methodology for the capacity constraints that takes advantage of our ability to solve small instances of the TSP efficiently. Specifically, when standard procedures fail to separate a candidate point, we attempt to decompose it into a convex combination of TSP tours; if successful, the tours present in this decomposition are examined for violated capacity constraints; if not, the Farkas Theorem provides a hyperplane separating the point from the TSP polytope. We present some extensions of this basic concept and a general framework within which it can be applied to other combinatorial models. Computational results are given for an implementation within the parallel branch, cut, and price framework SYMPHONY.

Hermite normal form computation using modulo determinant arithmetic

This paper describes a new class of Hermite normal form solution procedures which perform modulo determinant arithmetic throughout the computation. This class of procedures is shown to possess a polynomial time complexity bound which is a function of the length of the input string. Computational results are also given.

On stable set polyhedra for K1,3-free graphs

Abstract We give several classes of facets for the convex hull of incidence vectors of stable sets in a K 1,3 -free graph, including facets with ( a, a + 1)-valued coefficients, where a = 1, 2, 3,…. These provide counterexamples to three recent conjectures concerning such facets. We also give a necessary and sufficient condition for a minimal imperfect graph to be an odd hole or an odd antihole and indicate that minimal imperfect K 1,3 -free graphs satisfy the condition.

Line perfect graphs

The concept of line perfection of a graph is defined so that a simple graph is line perfect if and only if its line graph is perfect in the usual sense. Line perfect graphs are characterized as those which contain no odd cycles of size larger than 3. Two well-know theorems of Konig for bipartite graphs are shown to hold also for line perfect graphs; this extension provides a reinterpretation of the content of these theorems.The concept of line perfection of a graph is defined so that a simple graph is line perfect if and only if its line graph is perfect in the usual sense. Line perfect graphs are characterized as those which contain no odd cycles of size larger than 3. Two well-know theorems of König for bipartite graphs are shown to hold also for line perfect graphs; this extension provides a reinterpretation of the content of these theorems.

Stability Critical Graphs and Even Subdivisions of K4

A graph is stability critical (?-critical) if the removal of any edge increases the stability number. We give an affirmative answer to a question raised by Chvaktal, namely, that every connected, critical graph that is neither K2 nor an odd cycle contains an even subdivision of K4

Cardinality-restricted chains and antichains in partially ordered sets

Abstract For a given poset and positive integer κ, four problems are considered. Covering : Determine a minimum cardinality cover of the poset elements by chains (antichains), each of length (width) at most κ. Optimization : Given also weights on the poset elements, find a chain (antichain) of maximum total weight among those of length (width) at most κ. It is shown that the chain covering problem is NP-complete, while chain optimization is polynomial-time solvable. Several classes of facets are derived for the polytope generated by incidence vectors of antichains of width at most κ. Certain of these facets are then used to develop a polyhedral combinatorial algorithm for the antichain optimization problem. Computational results are given for the algorithm on randomly generated posets with up to 1005 elements and 4 ⩽ κ ⩽ 30.

Stability critical graphs and ranks facets of the stable set polytope

Abstract Rank inequalities due to stability critical (a-critical) graphs are used to develop a finite nested sequence of linear relaxations of the stable set polytope, the strongest of which provides an integral max-min relation: In a simple graph, the maximum size of a stable set is equal to the minimum (weighted) value of a cover of nodes by a-critical subgraphs. For a simple graph containing no even subdivision of K4, these results imply that every rank facet is due either to an edge or to an odd cycle; consequently, the max-min relation specializes to give that the cardinality of a largest stable set equals the minimum value of a node covering by edges and odd cycles. This leads to a polynomial-time algorithm to find a maximum stable set and a minimum valued cover of nodes by edges and odd cycles in such a graph.

Finite checkability for integer rounding properties in combinatorial programming problems

LetA be a nonnegative integral matrix with no zero columns. Theinteger round-up property holds forA if for each nonnegative integral vectorw, the solution value to the integer programming problem min{1 ⋅y: yA ≥ w, y ≥ 0, y integer} is obtained by rounding up to the nearest integer the solution value to the corresponding linear programming problem min{1 ⋅y: yA ≥ w, y ≥ 0}. Theinteger round-down property is similarly defined for a nonnegative integral matrixB with no zero rows by considering max{1 ⋅y: yB ≤ w, y ≥ 0, y integer} and its linear programming correspondent. It is shown that the integer round-up and round-down properties can be checked through a finite process. The method of proof motivates a new and elementary proof of Fulkersons Pluperfect Graph Theorem.

An abstract linear duality model

We investigate an abstract linear duality model which has as special instances several duality systems of interest in combinatorial optimization: subspace orthogonality, cone polarity, lattice duality, blocking polyhedra and antiblocking polyhedra. The descriptive duality present in the model is that of specifying a set in terms of linear constraints or viewing a set as being generated by certain types of linear combinations. We define properties of Weyl, Farkas and Minkowski for the general model by analogy with classical results in cone polarity and we investigate relationships among these properties and further properties of Lehman and Fulkerson, defined by analogy with results on dual pairs of polyhedra of the blocking or antiblocking type. In particular, we show that, for any given duality system, the Weyl property is equivalent to the combined properties of Farkas and Minkowski (or Lehman), that the Farkas property implies the Fulkerson property and that the Minkowski property is equivalent to the combined properties of Lehman and Fulkerson. We also adapt results of Dixon (Dixon, J. D. 1981. On duality theorems. Linear Algebra Appl. 39 223--228.) in order to obtain general sufficient conditions for validity of the Weyl property.In the second part of the paper, specific instances of the model are examined in more detail. The instances studied here are “integral duality” and “nonnegative integral duality,” which are related to the problem of finding integral solutions and nonnegative integral solutions, respectively, for linear systems. For each instance, the validity of the Weyl, Minkowski, Farkas, Lehman and Fulkerson properties is examined. We also characterize the constrained sets under each duality.

Packing and covering with integral feasible flows integral supply-demand networks

Polynomial-time algorithms are presented for solving combinatorial packing and covering problems defined from the integral feasible flows in an integral supply-demand network. These algorithms are also shown to apply to packing and covering problems defined by the minimal integral solutions to general totally unimodular systems. Analogous problems arising from matroid bases are also discussed and it is demonstrated that a means for solving such problems is provided by recent work of Cunningham.