An integer analogue of Carathéodory's theorem

Abstract

We prove a theorem on Hilbert bases analogous to Caratheodory's theorem for convex cones. The result is used to give an upper bound on the number of nonzero variables needed in optimal solutions to integer programs associated with totally dual integral systems. For integer programs arising from perfect graphs the general bounds are improved to show that if G is a perfect graph with n nodes and ''" is a vector of integral node weights, then there exists a minimum 11•-covering of the nodes that uses at most n distinct cliques. <" 1986 Academic Press, Inc I. INTRODUCTION A rational cone is a set of the form {x: Ax~O}, where A is a rational m x n matrix and 0 is the m component zero vector. By the theorems of Wey! and Minkowski (see [ 19]), C is a rational cone if and only if there exists a finite set of rational vectors {a 1 , ••• , ak} that generate C, that is, C = P, 1 a 1 +-• + },k ak:),;~ 0, i = !, ... , k }. The dimension of a rational cone C, denoted by dim C, is the cardinality of a maximal set of linearly independent vectors in C. A well-known result of Caratheodory 1s the following.

Integer Decomposition for Polyhedra Defined by Nearly Totally Unimodular Matrices

“…Then |a T x i − a T x j | ≤ 1 for any i and j, since otherwise we can replace x i and x j by λx i + (1 − λ)x j and λx j +(1−λ)x i , where λ = 1/|a T x i −a T x j | thus reducing (6). Hence a T x i ∈ {q, q +1} for each i = 1, .…”

“…With a little more care (as was pointed out by an anonymous referee), a decomposition can be found in polynomial time as follows (see [6,8]). First, t ≤ n + 1 affinely independent integer vectors y 1 , .…”

Alternatives for testing total dual integrality

“…All essential ingredients except unimodularity are already included in the proof of Fulkerson's pluperfect graph theorem [17]. Fulkerson's proof suggests a greedy way of taking active rows with an integer coefficient (see below); this is often exploited to prove that some particular systems are TDI, let us only cite two papers the closest to ours, Cook, Fonlupt & Schrijver [6] and Chandrasekaran & Tamir [3]. The latter paper extensively used lexicographically best solutions, which is an important tool in linear programming theory.…”

“…More precisely, the main result of Section 3 is the following: The proof relies on the basic idea of Fulkerson's famous "pluperfect graph theorem" [17] stating that the integrality of such polyhedra implies their total dual integrality in a very simple "greedy" way. Chandrasekaran and Tamir [3] and Cook, Fonlupt and Schrijver [6] exploited Fulkerson's method by pointing out its lexicographic or advantageous Carathéodory feature. In [35, §4] it is noticed with the same method that the active rows of the dual of integral set packing polyhedra (the cells of their normal fan) have a unimodular subdivision, which can be rephrased as follows: the normal fan of integral set packing polyhedra has a unimodular refinement.…”

Carathéodory bounds for integer cones

“…A finite set of vectors X is called a Hilbert basis, if each integer vector b ∈ cone(X) is a nonnegative integer combination of elements in X. Cook, Fonlupt and Schrijver [2] provided the following integer analogue of Carathéodory's theorem: If cone(X) is pointed and if X ⊆ Z d is an integral Hilbert basis, then for every b ∈ cone(X) there exists a subset X ⊆ X with | X| ≤ 2 d − 1 such that b is an integer conic combination of vectors in X. Sebő [8] improved this bound to 2 d − 2. Bruns et al [1] have shown that the bound d is not valid.…”