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Register a new userFocused Stochastic Local Search and the Lovasz Local Lemma
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Dimitris Achlioptas
Publication date
2015
fields
Informatics Engineering
and research's language is
English
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We develop tools for analyzing focused stochastic local search algorithms. These are algorithms which search a state space probabilistically by repeatedly selecting a constraint that is violated in the current state and moving to a random nearby state which, hopefully, addresses the violation without introducing many new ones. A large class of such algorithms arise from the algorithmization of the Lovasz Local Lemma, a non-constructive tool for proving the existence of satisfying states. Here we give tools that provide a unified analysis of such algorithms and of many more, expressing them as instances of a general framework.
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We develop a framework for the rigorous analysis of focused stochastic local search algorithms. These are algorithms that search a state space by repeatedly selecting some constraint that is violated in the current state and moving to a random nearby state that addresses the violation, while hopefully not introducing many new ones. An important class of focused local search algorithms with provable performance guarantees has recently arisen from algorithmizations of the Lov{a}sz Local Lemma (LLL), a non-constructive tool for proving the existence of satisfying states by introducing a background measure on the state space. While powerful, the state transitions of algorithms in this class must be, in a precise sense, perfectly compatible with the background measure. In many applications this is a very restrictive requirement and one needs to step outside the class. Here we introduce the notion of emph{measure distortion} and develop a framework for analyzing arbitrary focused stochastic local search algorithms, recovering LLL algorithmizations as the special case of no distortion. Our framework takes as input an arbitrary such algorithm and an arbitrary probability measure and shows how to use the measure as a yardstick of algorithmic progress, even for algorithms designed independently of the measure.
Let $Phi = (V, mathcal{C})$ be a constraint satisfaction problem on variables $v_1,dots, v_n$ such that each constraint depends on at most $k$ variables and such that each variable assumes values in an alphabet of size at most $[q]$. Suppose that each constraint shares variables with at most $Delta$ constraints and that each constraint is violated with probability at most $p$ (under the product measure on its variables). We show that for $k, q = O(1)$, there is a deterministic, polynomial time algorithm to approximately count the number of satisfying assignments and a randomized, polynomial time algorithm to sample from approximately the uniform distribution on satisfying assignments, provided that [Ccdot q^{3}cdot k cdot p cdot Delta^{7} < 1, quad text{where }C text{ is an absolute constant.}] Previously, a result of this form was known essentially only in the special case when each constraint is violated by exactly one assignment to its variables. For the special case of $k$-CNF formulas, the term $Delta^{7}$ improves the previously best known $Delta^{60}$ for deterministic algorithms [Moitra, J.ACM, 2019] and $Delta^{13}$ for randomized algorithms [Feng et al., arXiv, 2020]. For the special case of properly $q$-coloring $k$-uniform hypergraphs, the term $Delta^{7}$ improves the previously best known $Delta^{14}$ for deterministic algorithms [Guo et al., SICOMP, 2019] and $Delta^{9}$ for randomized algorithms [Feng et al., arXiv, 2020].
We study the problem of sampling an approximately uniformly random satisfying assignment for atomic constraint satisfaction problems i.e. where each constraint is violated by only one assignment to its variables. Let $p$ denote the maximum probability of violation of any constraint and let $Delta$ denote the maximum degree of the line graph of the constraints. Our main result is a nearly-linear (in the number of variables) time algorithm for this problem, which is valid in a Lovasz local lemma type regime that is considerably less restrictive compared to previous works. In particular, we provide sampling algorithms for the uniform distribution on: (1) $q$-colorings of $k$-uniform hypergraphs with $Delta lesssim q^{(k-4)/3 + o_{q}(1)}.$ The exponent $1/3$ improves the previously best-known $1/7$ in the case $q, Delta = O(1)$ [Jain, Pham, Vuong; arXiv, 2020] and $1/9$ in the general case [Feng, He, Yin; STOC 2021]. (2) Satisfying assignments of Boolean $k$-CNF formulas with $Delta lesssim 2^{k/5.741}.$ The constant $5.741$ in the exponent improves the previously best-known $7$ in the case $k = O(1)$ [Jain, Pham, Vuong; arXiv, 2020] and $13$ in the general case [Feng, He, Yin; STOC 2021]. (3) Satisfying assignments of general atomic constraint satisfaction problems with $pcdot Delta^{7.043} lesssim 1.$ The constant $7.043$ improves upon the previously best-known constant of $350$ [Feng, He, Yin; STOC 2021]. At the heart of our analysis is a novel information-percolation type argument for showing the rapid mixing of the Glauber dynamics for a carefully constructed projection of the uniform distribution on satisfying assignments. Notably, there is no natural partial order on the space, and we believe that the techniques developed for the analysis may be of independent interest.
Following the groundbreaking algorithm of Moser and Tardos for the Lovasz Local Lemma (LLL), there has been a plethora of results analyzing local search algorithms for various constraint satisfaction problems. The algorithms considered fall into two broad categories: resampling algorithms, analyzed via different algorithmic LLL conditions; and backtracking algorithms, analyzed via entropy compression arguments. This paper introduces a new convergence condition that seamlessly handles resampling, backtracking, and hybrid algorithms, i.e., algorithms that perform both resampling and backtracking steps. Unlike all past LLL work, our condition replaces the notion of a dependency or causality graph by quantifying point-to-set correlations between bad events. As a result, our condition simultaneously: (i)~captures the most general algorithmic LLL condition known as a special case; (ii)~significantly simplifies the analysis of entropy compression applications; (iii)~relates backtracking algorithms, which are conceptually very different from resampling algorithms, to the LLL; and most importantly (iv)~allows for the analysis of hybrid algorithms, which were outside the scope of previous techniques. We give several applications of our condition, including a new hybrid vertex coloring algorithm that extends the recent breakthrough result of Molloy for coloring triangle-free graphs to arbitrary graphs.
We consider the task of designing Local Computation Algorithms (LCA) for applications of the Lov{a}sz Local Lemma (LLL). LCA is a class of sublinear algorithms proposed by Rubinfeld et al.~cite{Ronitt} that have received a lot of attention in recent years. The LLL is an existential, sufficient condition for a collection of sets to have non-empty intersection (in applications, often, each set comprises all objects having a certain property). The ground-breaking algorithm of Moser and Tardos~cite{MT} made the LLL fully constructive, following earlier results by Beck~cite{beck_lll} and Alon~cite{alon_lll} giving algorithms under significantly stronger LLL-like conditions. LCAs under those stronger conditions were given in~cite{Ronitt}, where it was asked if the Moser-Tardos algorithm can be used to design LCAs under the standard LLL condition. The main contribution of this paper is to answer this question affirmatively. In fact, our techniques yield LCAs for settings beyond the standard LLL condition.
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