A Landscape Analysis of Constraint Satisfaction Problems

TL;DR

This paper analyzes constraint satisfaction problems through the lens of energy landscapes, revealing geometric properties and proposing algorithms that solve problems efficiently up to certain phase transitions, with implications for understanding problem hardness.

Contribution

It introduces a geometric framework for analyzing CSPs using energy landscapes, connecting solution space structure to algorithmic complexity and phase transitions.

Findings

Benchmark algorithm solves problems up to clustering transition

Geometric meaning of the easy-hard transition point

Better characterization of the J-point in packing problems

Abstract

We discuss an analysis of Constraint Satisfaction problems, such as Sphere Packing, K-SAT and Graph Coloring, in terms of an effective energy landscape. Several intriguing geometrical properties of the solution space become in this light familiar in terms of the well-studied ones of rugged (glassy) energy landscapes. A `benchmark' algorithm naturally suggested by this construction finds solutions in polynomial time up to a point beyond the `clustering' and in some cases even the `thermodynamic' transitions. This point has a simple geometric meaning and can be in principle determined with standard Statistical Mechanical methods, thus pushing the analytic bound up to which problems are guaranteed to be easy. We illustrate this for the graph three and four-coloring problem. For Packing problems the present discussion allows to better characterize the `J-point', proposed as a systematic definition of Random Close Packing, and to place it in the context of other theories of glasses.