The Graver Complexity of Integer Programming

Abstract

In this article we establish an exponential lower bound on the Graver complexity of integer programs. This provides new type of evidence supporting the presumable intractability of integer programming. Specifically, we show that the Graver complexity of the incidence matrix of the complete bipartite graph K 3, m satisfies g(m) = Ω (2 m ), with g(m) ≥ 17 · 2 m−3 − 7 for every m > 3.

A lower bound for the Graver complexity of the incidence matrix of a complete bipartite graph

“…Our result is a generalization of the result by Berstein and Onn [2] for the complete bipartite graph K 3,r , r ≥ 3. …”

“…In particular, from the universality of the three-way transportation program to general integer programs [3,6], the Graver complexity of the incidence matrix of the complete bipartite graph K 3,r is particularly important. Berstein and Onn [2] proved that the Graver complexity g(r) for the incidence matrix of K 3,r , r ≥ 3 is bounded below as g(r) = Ω(2 r ), where g(r) ≥ 17 · 2 r−3 − 7. It is a natural question to generalize this result to the complete bipartite graph K t,r of arbitrary size t, r. We prove that the Graver complexity for K t,r is Ω((t − 1) r ), where t ≥ 4 is fixed and r diverges to infinity.…”

“…The Graver complexity of an integer matrix is currently actively investigated for its importance to integer programming, algebraic statistics and other applications [2,4,5,1]. In particular, from the universality of the three-way transportation program to general integer programs [3,6], the Graver complexity of the incidence matrix of the complete bipartite graph K 3,r is particularly important.…”

“…It is a natural question to generalize this result to the complete bipartite graph K t,r of arbitrary size t, r. We prove that the Graver complexity for K t,r is Ω((t − 1) r ), where t ≥ 4 is fixed and r diverges to infinity. For proving our result, we employ double induction on r and t starting from the result of [2].…”

“…In this section, we summarize our notation and review relevant known results on the Graver complexity following [2].…”