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

2025-07-04

AI Summary

This work addresses the combinatorial $n$-fold integer linear programming (ILP) problem by proposing the first direct algorithm that bypasses both augmentation frameworks and linear programming (LP) relaxations. The core methodological innovation lies in the first application of the Steinitz Lemma to $n$-fold ILP, enabling direct construction of an optimal solution path via vector rearrangement analysis, dynamic programming, and geometric parameter control. The algorithm handles nonnegative unbounded integer variables, thereby overcoming the longstanding reliance of existing methods on iterative augmentation and LP-based preprocessing. Theoretical analysis shows that its running time either matches or improves upon the current state-of-the-art; for certain structured inputs, it achieves significant speedups. Its asymptotic improvement is equivalent to the concurrent, independent parallel work of Rohwedder. Overall, this work establishes a simpler, more intrinsic algorithmic paradigm for $n$-fold ILP.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We present an algorithm for a class of $n$-fold ILPs: whose existing algorithms in literature typically (1) are based on the extit{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.

Problem

• Solves Combinatorial n-fold ILPs using Steinitz Lemma
• Improves running time for certain input structures
• Avoids linear relaxation and augmentation framework

Innovation

• Simple algorithm for n-fold ILPs
• Uses Steinitz Lemma for solution
• Improves running time efficiency