Vectors in a Box

For an integer d ≥ 1, let τ(d) be the smallest integer with the following property: If v1,v2,...,vt is a sequence of t ≥ 2 vectors in [−1, 1]d with v1+v2+ · · ·+vt ∈ [−1, 1]d, then there is a set S ⊆ {1, 2,..., t} of indices, 2 ≤ |S | ≤ τ(d), such that ∑i∈S vi ∈ [−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(2d), 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) ≤ dd+o(d), and based on a construction of Alon and Vũ, whose main idea goes back to H̊astad, we obtain a lower bound of τ(d) ≥ dd/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