A New Perspective on Stochastic Local Search and the Lovasz Local Lemma

by   Dimitris Achlioptas, et al.

We present a new perspective on the analysis of stochastic local search algorithms, via linear algebra, and use it to establish a new criterion for their convergence. Our criterion captures and unifies the analysis of all currently known LLL-inspired local search algorithms, including all current applications of the entropy compression method. It can be seen as a generalization of the Lovasz Local Lemma that quantifies the interaction strength of bad events, so that weak interactions form correspondingly small obstacles to algorithmic convergence. As a demonstration of its power, we use our criterion to analyze a complex local search algorithm for the classical problem of coloring graphs with sparse neighborhoods. We prove that any improvement over our algorithm would require a major (and unexpected) breakthrough in random graph theory, suggesting that our criterion reaches the edge of tractability for this problem. Finally, we consider questions such as the number of possible distinct final states and the probability that certain portions of the state space are visited by a local search algorithm. Such information is currently available for the Moser-Tardos algorithm and for algorithms satisfying a combinatorial notion of commutativity introduced of Kolmogorov. Our framework provides a very natural and more general notion of commutativity (essentially matrix commutativity) which allows the recovery of all such results with much simpler proofs.


page 1

page 2

page 3

page 4


A new notion of commutativity for the algorithmic Lovász Local Lemma

The Lovász Local Lemma (LLL) is a powerful tool in probabilistic combina...

A Local Lemma for Focused Stochastic Algorithms

We develop a framework for the rigorous analysis of focused stochastic l...

A Stochastic Process Model of Classical Search

Among classical search algorithms with the same heuristic information, w...

Unweighted Stochastic Local Search can be Effective for Random CSP Benchmarks

We present ULSA, a novel stochastic local search algorithm for random bi...

Algorithms for Optimizing Fleet Staging of Air Ambulances

In a disaster situation, air ambulance rapid response will often be the ...

Crystal Structure Prediction via Oblivious Local Search

We study Crystal Structure Prediction, one of the major problems in comp...

Mixed-model Sequencing with Reinsertion of Failed Vehicles: A Case Study for Automobile Industry

In the automotive industry, some vehicles, failed vehicles, cannot be pr...

Please sign up or login with your details

Forgot password? Click here to reset