The Graver Complexity of Integer Programming

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