In the classic Integer Programming (IP) problem, the objective is to decide
whether, for a given m×n matrix A and an m-vector b=(b1,…,bm), 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.Comment: 29 pages, To appear in ESA 201