On the optimality of pseudo-polynomial algorithms for integer programming

In the classic Integer Programming Feasibility (IPF) problem, the objective is to decide whether, for a given m × 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 (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 pseudo-polynomial 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(m )^m log ( ) log ( +‖ b‖ _∞)+𝒪(mn) . Here, 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 n^o(m/log m)·‖ b‖ _∞^o(m) . We also prove that assuming ETH, (IPF) cannot be solved in time f(m)· (n ·‖ b‖ _∞)^o(m/log m) 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 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 (IPF) 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 (IPF) when the path-width of the corresponding column-matroid is a constant .