Circuit walks in integral polyhedra

Abstract

Circuits play a fundamental role in the theory of linear programming due to their intimate connection to algorithms of combinatorial optimization and the efficiency of the simplex method. We are interested in better understanding the properties of circuit walks in integral polyhedra. In this paper, we introduce a hierarchy for integral polyhedra based on different types of behavior exhibited by their circuit walks. Many problems in combinatorial optimization fall into the most interesting categories of this hierarchy -steps of circuit walks only stop at integer points, at vertices, or follow actual edges. We classify several classical families of polyhedra within the hierarchy, including 0/1-polytopes, polyhedra defined by totally unimodular matrices, and more specifically matroid polytopes, transportation polytopes, and partition polytopes. Finally, we prove three characterizations of the simple polytopes that appear in the bottom level of the hierarchy where all circuit walks are edge walks, showing that such polytopes constitute a generalization of simplices and parallelotopes.

A polyhedral model for enumeration and optimization over the set of circuits

“…A consequence of Lemma 4 is that the number of iterations of the steepest-descent augmentation algorithm is bounded by m B times the number of different values of −c T g/||Bg|| 1 over all circuits g ∈ C(A, B). If A B is totally unimodular, it holds that g ∈ {0, 1, −1} n and Bg ∈ {0, 1, −1} mB for each circuit [8]. Hence, since |c T g| ≤ ||c|| 1 and ||Bg|| 1 ≤ m B , the number of augmentation steps is at most ||c|| 1 (m B ) 2 .…”

“…Actually constructing such a walk would yield a short sequence of transitions from v 1 to v 2 using only the circuits of P . As seen in [6,8], circuit walks in polyhedra from combinatorial optimization often have intuitive interpretations in terms of the underlying problem. Sign-compatible circuit walks exhibit additional desirable properties.…”

“…For instance, in a standard form polyhedron, a sign-compatible circuit walk only ever increases (decreases) each variable if it needs to be increased (decreased) to reach the destination v 2 . In the context of the partition polytopes of [4,8], for example, this means that any item is exchanged at most once when transitioning between given clusterings of a set.…”

“…We may wish to construct a sign-compatible circuit walk from v 1 to v 2 in which the objective function decreases as quickly as possible. For instance, when transitioning between the clusterings of a set given by the partition polytopes of [4,8], such a walk would correspond to a sequence of exchanges in which each individual exchange reduces the objective function by a maximum possible value per item moved. We will call such a circuit walk a c-steepest sign-compatible circuit walk.…”

“…In the context of combinatorial optimization, a circuit walk should be integral in order to have a natural interpretation in terms of the underlying problem. It is shown in [8] that a totally unimodular constraint matrix implies that any maximal circuit walk in the associated polyhedron is integral. However, a sign-compatible circuit walk constructed via Algorithm 4 need not be maximal.…”