A constructive algorithm for the LLL on permutations

TL;DR

This paper introduces a polynomial-time randomized algorithm for the Lovász Local Lemma in the context of permutations, enabling efficient construction of combinatorial objects like Latin transversals, which was previously unresolved.

Contribution

It develops the first polynomial-time algorithm for the LLL on permutations and extends the Moser-Tardos framework to this setting, with applications to Latin transversals and other combinatorial structures.

Findings

First polynomial-time algorithm for LLL on permutations

Constructs Latin transversals efficiently under new conditions

Develops RNC algorithms for various combinatorial problems

Abstract

While there has been significant progress on algorithmic aspects of the Lov\'{a}sz Local Lemma (LLL) in recent years, a noteworthy exception is when the LLL is used in the context of random permutations. The breakthrough algorithm of Moser & Tardos only works in the setting of independent variables, and does not apply in this context. We resolve this by developing a randomized polynomial-time algorithm for such applications. A noteworthy application is for Latin transversals: the best-known general result here (Bissacot et al., improving on Erd\H{o}s and Spencer), states that any $n \times n$ matrix in which each entry appears at most $(27/256)n$ times, has a Latin transversal. We present the first polynomial-time algorithm to construct such a transversal. We also develop RNC algorithms for Latin transversals, rainbow Hamiltonian cycles, strong chromatic number, and hypergraph packing. In addition to efficiently finding a configuration which avoids bad-events, the algorithm of Moser & Tardos has many powerful extensions and properties. These include a well-characterized distribution on the output distribution, parallel algorithms, and a partial resampling variant. We show that our algorithm has nearly all of the same useful properties as the Moser-Tardos algorithm, and present a comparison of this aspect with recent works on the LLL in general probability spaces.