Faster and simpler traversal of 0/1-polytopes

2026-07-20

AI Summary

This work addresses the problem of efficiently traversing Hamiltonian paths over the vertices of 0/1-polytopes. Relying solely on a linear optimization oracle, the authors propose a combinatorial traversal algorithm that reduces the amortized delay per vertex from O(t·log n) to O(t), where t denotes the time required for a single linear optimization call, thereby eliminating the logarithmic factor. The method applies broadly to 0/1-polytopes associated with various discrete structures—including matroid bases, graph matchings, vertex covers, and antichains in posets—significantly enhancing enumeration efficiency and improving the theoretical delay bound for vertex enumeration of 0/1-polytopes.

Abstract

Recently, Merino and Mütze (FOCS'23+SICOMP'24) presented an algorithm for computing a Hamilton path on the skeleton of any 0/1-polytope ${\rm conv}(X)$, where $X\subseteq\{0,1\}^n$. The algorithm uses as a black box an algorithm for solving the classical linear optimization problem $\min\{w\cdot x\mid x\in X\}$ for some weight vector $w\in\mathbb{R}^n$. The resulting delay per visited vertex on the Hamilton path is only by a $\log n$ factor larger than the time to solve one instance of the optimization algorithm. In this paper, we make the Hamilton path algorithm simpler and faster. Namely, we obtain an amortized delay that is only by a constant factor larger than the running time of the optimization algorithm, thus removing the $\log n$ factor. As concrete results, this yields improved algorithms for generating bases and independent sets in a matroid, spanning trees, forests, matchings and maximum matchings in a graph, vertex covers, minimum vertex covers, independent sets and maximum independent sets in a bipartite graph, and antichains, maximum antichains and ideals in a poset. All of these listings correspond to Hamilton paths on the corresponding polytopes. Furthermore, we obtain an $\mathcal{O}(t_{\rm LP})$ amortized delay algorithm for the vertex enumeration problem on 0/1-polytopes $\{x\in\mathbb{R}^n\mid Ax\leq b\}$, where $A\in \mathbb{R}^{m\times n}$ and $b\in\mathbb{R}^m$, and $t_{\rm LP}$ is the time needed to solve the linear program $\min\{w\cdot x\mid Ax\leq b\}$. This improves upon the $\mathcal{O}(t_{\rm LP} \log n)$ delay algorithm of Merino and Mütze, and the previous $\mathcal{O}(t_{\rm LP}\,n)$ delay algorithm of Bussieck and Lübbecke from 1998.

Problem

• 0/1-polytopes
• Hamilton path
• vertex enumeration
• combinatorial generation
• amortized delay

Innovation

• 0/1-polytopes
• Hamilton path
• amortized delay
• linear optimization
• vertex enumeration