A simple algorithm for Combinatorial n-fold ILPs using the Steinitz Lemma

TL;DR

This paper introduces a straightforward algorithm for combinatorial n-fold integer linear programs using the Steinitz lemma, offering improvements over existing methods in terms of simplicity and efficiency.

Contribution

The paper presents a novel, simple algorithm for combinatorial n-fold ILPs leveraging the Steinitz lemma, bypassing the need for linear relaxations and augmentation frameworks.

Findings

The algorithm effectively solves combinatorial n-fold ILPs with unbounded non-negative variables.

It improves the running time of existing algorithms depending on input structure.

The approach offers a more direct and simpler solution method.

Abstract

We present an algorithm for a class of $n$-fold ILPs: whose existing algorithms in literature typically (1) are based on the \textit{augmentation framework} where one starts with an arbitrary solution and then iteratively moves towards an optimal solution by solving appropriate programs; and (2) require solving a linear relaxation of the program. Combinatorial $n$-fold ILPs is a class introduced and studied by Knop et al. [MP2020] that captures several other problems in a variety of domains. We present a simple and direct algorithm that solves Combinatorial $n$-fold ILPs with unbounded non-negative variables via an application of the Steinitz lemma, a classic result regarding reordering of vectors. Depending on the structure of the input, we also improve upon the existing algorithms in literature in terms of the running time, thereby showing an improvement that mirrors the one shown by Rohwedder [ICALP2025] contemporaneously and independently.