On Integer Programming and Convolution

Integer programs with a fixed number of constraints can be solved in pseudo-polynomial time. We present a surprisingly simple algorithm and matching conditional lower bounds. Consider an IP in standard form max{cx : Ax = b, x ∈ Zn≥0}, where A ∈ Z , b ∈ Z and c ∈ Z. Let ∆ be an upper bound on the absolute values in A. We show that this IP can be solved in time O(m∆) · log(‖b‖∞). The previous best algorithm has a running time of n ·O(m∆) · ‖b‖1. The hardness of (min, +)-convolution has been used to prove conditional lower bounds on a number of polynomially solvable problems. We show that improving our algorithm for IPs of any fixed number of constraints is equivalent to improving (min, +)-convolution. More precisely, for any fixed m there exists an algorithm for solving IPs with m constraints in time O(m(∆ + ‖b‖∞)) 2m−δ for some δ > 0, if and only if there is a truly sub-quadratic algorithm for (min, +)-convolution. For the feasibility problem, where the IP has no objective function, we improve the running time to O(m∆) log(∆) log(∆+ ‖b‖∞). We also give a matching lower bound here: For every fixed m and δ > 0 there is no algorithm for testing feasibility of IPs with m constraints in time n ·O(m(∆+ ‖b‖∞)) m−δ unless the SETH is false.