The Graver Complexity of Integer Programming

TL;DR

This paper proves an exponential lower bound on the Graver complexity of certain integer programming problems, indicating their inherent computational difficulty and supporting the conjecture of their intractability.

Contribution

It establishes the first exponential lower bound on the Graver complexity of the incidence matrix of complete bipartite graphs, advancing understanding of integer program complexity.

Findings

Graver complexity of $K_{3,m}$ is at least exponential in m

Provides explicit lower bounds for Graver complexity

Supports the conjecture of integer programming intractability

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 $g(m)=\Omega(2^m)$, with $g(m)\geq 17\cdot 2^{m-3}-7$ for every $m>3$ .