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On the Optimality of Pseudo-polynomial Algorithms for Integer Programming Fomin, Fedor V. Panolan, Fahad Ramanujan, M. S. Saurabh, Saket Integer Programming Strong Exponential Time Hypothesis Branch-width of a matrix Fine-grained Complexity In the classic Integer Programming (IP) problem, the objective is to decide whether, for a given m x n matrix A and an m-vector b=(b_1,..., b_m), there is a non-negative integer n-vector x such that Ax=b. Solving (IP) is an important step in numerous algorithms and it is important to obtain an understanding of the precise complexity of this problem as a function of natural parameters of the input. The classic pseudo-polynomial time algorithm of Papadimitriou [J. ACM 1981] for instances of (IP) with a constant number of constraints was only recently improved upon by Eisenbrand and Weismantel [SODA 2018] and Jansen and Rohwedder [ArXiv 2018]. We continue this line of work and show that under the Exponential Time Hypothesis (ETH), the algorithm of Jansen and Rohwedder is nearly optimal. We also show that when the matrix A is assumed to be non-negative, a component of Papadimitriou's original algorithm is already nearly optimal under ETH. This motivates us to pick up the line of research initiated by Cunningham and Geelen [IPCO 2007] who studied the complexity of solving (IP) with non-negative matrices in which the number of constraints may be unbounded, but the branch-width of the column-matroid corresponding to the constraint matrix is a constant. We prove a lower bound on the complexity of solving (IP) for such instances and obtain optimal results with respect to a closely related parameter, path-width. Specifically, we prove matching upper and lower bounds for (IP) when the path-width of the corresponding column-matroid is a constant. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Fedor V. Fomin and Fahad Panolan and M. S. Ramanujan and Saket Saurabh 2018 Is Part Of LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ESA.2018.31 urn:nbn:de:0030-drops-94949 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.31 eng https://creativecommons.org/licenses/by/3.0/legalcode