Relationship between clustering and algorithmic phase transitions in the random k-XORSAT model and its NP-complete extensions

TL;DR

This paper investigates how clustering phenomena in random k-XORSAT models affect the success of heuristic algorithms, revealing that clustering limits the solvable ratio and that certain heuristics can reach this limit as k grows large.

Contribution

It establishes a relationship between clustering thresholds and heuristic performance in random CSPs, introducing bounds on solvability ratios and analyzing heuristic effectiveness.

Findings

Heuristic algorithms have a maximum ratio smaller than the clustering threshold.

Clustering occurs at a lower ratio than the one where heuristics typically succeed.

The Generalized Unit Clause heuristic can reach the clustering ratio as k increases.

Abstract

We study the performances of stochastic heuristic search algorithms on Uniquely Extendible Constraint Satisfaction Problems with random inputs. We show that, for any heuristic preserving the Poissonian nature of the underlying instance, the (heuristic-dependent) largest ratio $\alpha_a$ of constraints per variables for which a search algorithm is likely to find solutions is smaller than the critical ratio $\alpha_d$ above which solutions are clustered and highly correlated. In addition we show that the clustering ratio can be reached when the number k of variables per constraints goes to infinity by the so-called Generalized Unit Clause heuristic.