Statistical Physics of Hard Optimization Problems

TL;DR

This paper explores the complexity of NP-complete optimization problems using statistical physics methods, identifying properties that distinguish typically hard instances from easier ones, and introduces a new class of challenging problems called "locked" constraints.

Contribution

It applies the cavity method to analyze solution spaces of satisfiability and graph coloring problems, revealing links between frozen variables and computational hardness, and introduces "locked" problems as a new challenging class.

Findings

Solution space properties relate to problem hardness.

Frozen variables correlate with computational difficulty.

"Locked" problems are more challenging than traditional satisfiability.

Abstract

Optimization is fundamental in many areas of science, from computer science and information theory to engineering and statistical physics, as well as to biology or social sciences. It typically involves a large number of variables and a cost function depending on these variables. Optimization problems in the NP-complete class are particularly difficult, it is believed that the number of operations required to minimize the cost function is in the most difficult cases exponential in the system size. However, even in an NP-complete problem the practically arising instances might, in fact, be easy to solve. The principal question we address in this thesis is: How to recognize if an NP-complete constraint satisfaction problem is typically hard and what are the main reasons for this? We adopt approaches from the statistical physics of disordered systems, in particular the cavity method developed originally to describe glassy systems. We describe new properties of the space of solutions in two of the most studied constraint satisfaction problems - random satisfiability and random graph coloring. We suggest a relation between the existence of the so-called frozen variables and the algorithmic hardness of a problem. Based on these insights, we introduce a new class of problems which we named "locked" constraint satisfaction, where the statistical description is easily solvable, but from the algorithmic point of view they are even more challenging than the canonical satisfiability.