Constraint satisfaction problems with isolated solutions are hard

TL;DR

This paper investigates the phase diagram and computational hardness of locked constraint satisfaction problems, revealing that their isolated solutions make them mathematically tractable but algorithmically extremely challenging, serving as new benchmarks.

Contribution

It introduces locked CSPs with isolated solutions, analyzes their phase diagram, and demonstrates their high algorithmic hardness, contrasting with non-locked problems.

Findings

Locked problems have isolated solutions simplifying phase diagram analysis.

Algorithms fail to solve locked problems in the clustered phase.

Hardness transition coincides with the clustering transition in locked problems.

Abstract

We study the phase diagram and the algorithmic hardness of the random `locked' constraint satisfaction problems, and compare them to the commonly studied 'non-locked' problems like satisfiability of boolean formulas or graph coloring. The special property of the locked problems is that clusters of solutions are isolated points. This simplifies significantly the determination of the phase diagram, which makes the locked problems particularly appealing from the mathematical point of view. On the other hand we show empirically that the clustered phase of these problems is extremely hard from the algorithmic point of view: the best known algorithms all fail to find solutions. Our results suggest that the easy/hard transition (for currently known algorithms) in the locked problems coincides with the clustering transition. These should thus be regarded as new benchmarks of really hard constraint satisfaction problems.