Abstract
In the classic Integer Programming Feasibility (IPF) problem, the objective is to decide whether, for a given \(m \times n\) matrix A and an m-vector \(b=(b_1,\dots , b_m)\), 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 \({\mathcal {O}}(m \varDelta )^m\) \(\log (\varDelta ) \log (\varDelta +\Vert b\Vert _{\infty })+{\mathcal {O}}(mn)\). Here, \(\varDelta \) 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 \(n^{o(\frac{m}{\log m})} \cdot \Vert b\Vert _{\infty }^{o(m)}\). We also prove that assuming ETH, (IPF) cannot be solved in time \(f(m)\cdot (n \cdot \Vert b\Vert _{\infty })^{o(\frac{m}{\log m})}\) 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 .
Highlights
In the classic Integer Programming problem, the input is an m × n integer matrix A, and an m-vector b = (b1, . . . , bm)
We consider the feasibility version of the problem, where the objective is to find a non-negative integer n-vector x such that Ax = b. Solving this problem, denoted by (IP), is a fundamental 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
Unless Strong Exponential Time Hypothesis (SETH) fails, (IP) with even a non-negative m × n constraint matrix A cannot be solved in time f (k)( b ∞ + 1)(1− )k(mn)O(1) for any computable function f and
Summary
For (IP) with a non-negative matrix A, Cunningham and Geelen [1] showed that when the branch-width of the column-matroid of A is constant, (IP) is solvable in pseudo-polynomial time. Unless SETH fails, (IP) with even a non-negative m × n constraint matrix A cannot be solved in time f ( b ∞)( b ∞ + 1)(1− )k(mn)O(1) for any computable function f and > 0, where k is the path-width of the column matroid of A. (IP) with non-negative m×n matrix A given together with a path decomposition of its column matroid of width k is solvable in time O(( b ∞ + 1)k+1mn + m2n). A close inspection of the instances they construct in their NP-hardness reduction shows that the column matroids of the resulting constraint matrices are direct sums of circuits, implying that even their path-width is bounded by 3
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