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Space-Efficient Algorithm for Integer Programming with Few Constraints

Lars Rohwedder, Karol Węgrzycki

TL;DR

A polynomial space algorithm is presented 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.

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 $(mΔ)^{O(m)} \text{poly}(I)$, where $Δ$ 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 $(mΔ)^{O(m (\log m + \log\logΔ))} \text{poly}(I)$ time, that is, in almost the same time as previous dynamic programming algorithms.

Space-Efficient Algorithm for Integer Programming with Few Constraints

TL;DR

A polynomial space algorithm is presented that solves integer linear programs in time, that is, in almost the same time as previous dynamic programming algorithms.

Abstract

Integer linear programs , where , , and , can be solved in pseudopolynomial time for any fixed number of constraints . More precisely, in time , where is the maximum absolute value of an entry in and 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 time, that is, in almost the same time as previous dynamic programming algorithms.
Paper Structure (5 sections, 7 theorems, 7 equations, 1 algorithm)

This paper contains 5 sections, 7 theorems, 7 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Overview of previous and new techniques
  3. Branching algorithm
  4. Candidate support sets and main theorem
  5. Conclusion

Key Result

Theorem 1

Let $A \in \mathbb{Z}^{m \times n}$, $b \in \mathbb{Z}^m$, and $c \in \mathbb{Z}^n$ and suppose that the absolute value of each entry of $A$ is bounded by $\Delta > 1$. Then, in time and polynomial space, one can compute a solution to eq:ilp or conclude that such a solution does not exist. Here, $I$ is the encoding size of $A, b$, and $c$.

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2: cf. Lemma 3.2 in jin2021fast
  • Theorem 3: cf. Corollary 5 in caratheodory06
  • Theorem 4: cf. Theorem 5.2 in frank1987application
  • Lemma 5
  • proof : Proof of \ref{['lem:simple']}
  • Definition 6: Candidate support set
  • Lemma 7
  • proof : Proof of \ref{['thm:main']} assuming \ref{['lem:candidates']}
  • Lemma 8
  • ...and 2 more