Fomin, Fedor V. and Panolan, Fahad and Ramanujan, M. S. and et al, .
(2022)
On the optimality of pseudopolynomial algorithms for integer programming.
Mathematical Programming.
ISSN 00255610
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Abstract
In the classic Integer Programming Feasibility (IPF) problem, the objective is to decide whether, for a given m× n matrix A and an mvector b= (b1, ⋯ , bm) , there is a nonnegative integer nvector x such that Ax= b. Solving (IPF) 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 pseudopolynomial time algorithm of Papadimitriou [J. ACM 1981] for instances of (IPF) with a constant number of constraints was only recently improved upon by Eisenbrand and Weismantel [SODA 2018] and Jansen and Rohwedder [ITCS 2019]. Jansen and Rohwedder designed an algorithm for (IPF) with running time O(mΔ) mlog (Δ) log (Δ+ ‖ b‖ ∞) + O(mn). Here, Δ is an upper bound on the absolute values of the entries of A. We continue this line of work and show that under the Exponential Time Hypothesis (ETH), the algorithm of Jansen and Rohwedder is nearly optimal, by proving a lower bound of no(mlogm)·‖b‖∞o(m). We also prove that assuming ETH, (IPF) cannot be solved in time f(m)·(n·‖b‖∞)o(mlogm) for any computable function f. This motivates us to pick up the line of research initiated by Cunningham and Geelen [IPCO 2007] who studied the complexity of solving (IPF) with nonnegative matrices in which the number of constraints may be unbounded, but the branchwidth of the columnmatroid corresponding to the constraint matrix is a constant. We prove a lower bound on the complexity of solving (IPF) for such instances and obtain optimal results with respect to a closely related parameter, pathwidth. Specifically, we prove matching upper and lower bounds for (IPF) when the pathwidth of the corresponding columnmatroid is a constant. © 2022, SpringerVerlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society.
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