Dynamic Construction of the Lov\'asz Local Lemma

TL;DR

This paper extends local search algorithms for the Lovász Local Lemma to fully dynamic settings with an adaptive adversary, achieving near-linear total resampling steps per update and applications like efficient edge coloring.

Contribution

It proves that simple local search procedures can be efficiently maintained dynamically, even against adaptive adversaries, with near-linear total resampling steps.

Findings

Achieves $ ilde{O}(1)$ amortized local search steps per update in dynamic settings.

Extends convergence guarantees of local search algorithms to fully dynamic adversarial models.

Provides applications such as maintaining approximate edge colorings with polylogarithmic update time.

Abstract

This paper proves that a wide class of local search algorithms extend as is to the fully dynamic setting with an adaptive adversary, achieving an amortized $\tilde{O}(1)$ number of local-search steps per update. A breakthrough by Moser (2009) introduced the witness-tree and entropy compression techniques for analyzing local resampling processes for the Lov\'asz Local Lemma. These methods have since been generalized and expanded to analyze a wide variety of local search algorithms that can efficiently find solutions to many important local constraint satisfaction problems. These algorithms either extend a partial valid assignment and backtrack by unassigning variables when constraints become violated, or they iteratively fix violated constraints by resampling their variables. These local resampling or backtracking procedures are incredibly flexible, practical, and simple to specify and implement. Yet, they can be shown to be extremely efficient on static instances, typically performing only (sub)-linear number of fixing steps. The main technical challenge lies in proving conditions that guarantee such rapid convergence. This paper extends these convergence results to fully dynamic settings, where an adaptive adversary may add or remove constraints. We prove that applying the same simple local search procedures to fix old or newly introduced violations leads to a total number of resampling steps near-linear in the number of adversarial updates. Our result is very general and yields several immediate corollaries. For example, letting $\Delta$ denote the maximum degree, for a constant $\epsilon$ and $\Delta = \text{poly}(\log n)$, we can maintain a $(1+\epsilon) \Delta$-edge coloring in $\text{poly}(\log n)$ amortized update time against an adaptive adversary. The prior work for this regime has exponential running time in $\sqrt{\log n}$ [Christiansen, SODA '26].