On Integer Programming, Discrepancy, and Convolution

Integer programs with m constraints are solvable in pseudo-polynomial time in $Δ$, the largest coefficient in a constraint, when m is a fixed constant. We give a new algorithm with a running time of $O(\sqrt{m}Δ)^{2m} + O(nm)$, which improves on the state-of-the-art. Moreover, we show that improving on our algorithm for any $m$ is equivalent to improving over the quadratic time algorithm for $(\min,~+)$-convolution. This is a strong evidence that our algorithm's running time is the best possible. We also present a specialized algorithm with running time $O(\sqrt{m} Δ)^{(1 + o(1))m} + O(nm)$ for testing feasibility of an integer program and also give a tight lower bound, which is based on the SETH in this case.

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