Vectors in a box

Abstract

For an integer d ≥ 1, let τ(d) be the smallest integer with the following property: if v ₁, v ₂, . . . , v _t is a sequence of t ≥ 2 vectors in [−1, 1]^d with ${{\bf v}_1+{\bf v}_2+\cdots+{\bf v}_t \in [-1,1]^d}$ , then there is a set ${S\subseteq \{1,2,\ldots,t\}}$ of indices, 2 ≤ |S| ≤ τ(d), such that ${\sum_{i \in S}{\bf v}_i \in [-1,1]^d}$ . The quantity τ(d) was introduced by Dash, Fukasawa, and Günlük, who showed that τ(2) = 2, τ(3) = 4, and τ(d) = Ω(2^d), and asked whether τ(d) is finite for all d. Using the Steinitz lemma, in a quantitative version due to Grinberg and Sevastyanov, we prove an upper bound of τ(d) ≤ d ^d+o(d), and based on a construction of Alon and Vũ, whose main idea goes back to Håstad, we obtain a lower bound of τ(d) ≥ d ^d/2-o(d). These results contribute to understanding the master equality polyhedron with multiple rows defined by Dash et al. which is a “universal” polyhedron encoding valid cutting planes for integer programs (this line of research was started by Gomory in the late 1960s). In particular, the upper bound on τ(d) implies a pseudo-polynomial running time for an algorithm of Dash et al. for integer programming with a fixed number of constraints. The algorithm consists in solving a linear program, and it provides an alternative to a 1981 dynamic programming algorithm of Papadimitriou.