Space-Efficient Algorithm for Integer Programming with Few Constraints

TL;DR

This paper introduces a space-efficient algorithm for solving fixed-constraint integer linear programs, achieving near-equivalent time complexity to existing methods but with significantly reduced space requirements.

Contribution

The authors develop a polynomial space algorithm for fixed-constraint integer programming that maintains nearly the same time complexity as traditional dynamic programming approaches.

Findings

Achieves polynomial space complexity for fixed-constraint ILP.

Maintains almost the same time complexity as dynamic programming algorithms.

Provides a practical approach for large instances with limited memory.

Abstract

Integer linear programs $\min\{c^T x : A x = b, x \in \mathbb{Z}^n_{\ge 0}\}$, where $A \in \mathbb{Z}^{m \times n}$, $b \in \mathbb{Z}^m$, and $c \in \mathbb{Z}^n$, can be solved in pseudopolynomial time for any fixed number of constraints $m = O(1)$. More precisely, in time $(m\Delta)^{O(m)} \text{poly}(I)$, where $\Delta$ is the maximum absolute value of an entry in $A$ and $I$ the input size. Known algorithms rely heavily on dynamic programming, which leads to a space complexity of similar order of magnitude as the running time. In this paper, we present a polynomial space algorithm that solves integer linear programs in $(m\Delta)^{O(m (\log m + \log\log\Delta))} \text{poly}(I)$ time, that is, in almost the same time as previous dynamic programming algorithms.