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Polynomial Time Algorithms for Integer Programming and Unbounded Subset Sum in the Total Regime

Divesh Aggarwal, Antoine Joux, Miklos Santha, Karol Węgrzycki

TL;DR

It is shown that when $b$ is slightly greater than the Frobenius number, the solution to USS can be found in polynomial time, and how this results extend to Integer Programming with Equalities (ILPE), highlighting conditions under which ILPE becomes total.

Abstract

The Unbounded Subset Sum (USS) problem is an NP-hard computational problem where the goal is to decide whether there exist non-negative integers $x_1, \ldots, x_n$ such that $x_1 a_1 + \ldots + x_n a_n = b$, where $a_1 < \cdots < a_n < b$ are distinct positive integers with $\text{gcd}(a_1, \ldots, a_n)$ dividing $b$. The problem can be solved in pseudopolynomial time, while specialized cases, such as when $b$ exceeds the Frobenius number of $a_1, \ldots, a_n$ simplify to a total problem where a solution always exists. This paper explores the concept of totality in USS. The challenge in this setting is to actually find a solution, even though we know its existence is guaranteed. We focus on the instances of USS where solutions are guaranteed for large $b$. We show that when $b$ is slightly greater than the Frobenius number, we can find the solution to USS in polynomial time. We then show how our results extend to Integer Programming with Equalities (ILPE), highlighting conditions under which ILPE becomes total. We investigate the diagonal Frobenius number, which is the appropriate generalization of the Frobenius number to this context. In this setting, we give a polynomial-time algorithm to find a solution of ILPE. The bound obtained from our algorithmic procedure for finding a solution almost matches the recent existential bound of Bach, Eisenbrand, Rothvoss, and Weismantel (2024).

Polynomial Time Algorithms for Integer Programming and Unbounded Subset Sum in the Total Regime

TL;DR

It is shown that when is slightly greater than the Frobenius number, the solution to USS can be found in polynomial time, and how this results extend to Integer Programming with Equalities (ILPE), highlighting conditions under which ILPE becomes total.

Abstract

The Unbounded Subset Sum (USS) problem is an NP-hard computational problem where the goal is to decide whether there exist non-negative integers such that , where are distinct positive integers with dividing . The problem can be solved in pseudopolynomial time, while specialized cases, such as when exceeds the Frobenius number of simplify to a total problem where a solution always exists. This paper explores the concept of totality in USS. The challenge in this setting is to actually find a solution, even though we know its existence is guaranteed. We focus on the instances of USS where solutions are guaranteed for large . We show that when is slightly greater than the Frobenius number, we can find the solution to USS in polynomial time. We then show how our results extend to Integer Programming with Equalities (ILPE), highlighting conditions under which ILPE becomes total. We investigate the diagonal Frobenius number, which is the appropriate generalization of the Frobenius number to this context. In this setting, we give a polynomial-time algorithm to find a solution of ILPE. The bound obtained from our algorithmic procedure for finding a solution almost matches the recent existential bound of Bach, Eisenbrand, Rothvoss, and Weismantel (2024).
Paper Structure (6 sections, 14 theorems, 26 equations)

This paper contains 6 sections, 14 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Preliminaries
  4. Algorithm for Unbounded Subset Sum
  5. Algorithm for Solving Variants of the ILP
  6. Lower Bound for the Totality Condition for ILP with Equalities

Key Result

Theorem 1.1

Let $k$ be a non-negative integer. There is a $\mathrm{poly}(n,\log b) \cdot (\log k)^{O(k)}$ time algorithm that given an Unbounded Subset Sum instance $(n, a_1, \ldots, a_n, b)$ such that $b \ge \frac{a_i^2}{i-1}$, for all $k < i \leq n$, and $\gcd(a_1, \ldots, a_n)$ divides $b$, finds $x_1, \ldot

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2: Weaker version of Theorem \ref{['thm:pre-main']}
  • Theorem 1.3: Theorem \ref{['thm:counter-example']} simplified
  • Definition 2.1: Integer Linear Programming (ILP)
  • Definition 2.2: Integer Linear Programming with Equalities (ILPE)
  • Definition 2.3: Unbounded Subset Sum (USS)
  • Definition 2.4: Heterogeneous Integer Linear Programming (HILP)
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • ...and 18 more