Detecting the finer structure of the P vs NP problem with statistical mechanics: the case of the Wang tiling problem

TL;DR

This paper explores the P vs NP problem's nuanced structure using statistical mechanics, focusing on the Wang tiling problem to distinguish regions where polynomial algorithms are feasible or chaotic, revealing a complex landscape.

Contribution

It introduces a physics-inspired heuristic to classify problem instances into regions with different computational behaviors, providing new insights into NP-complete problems.

Findings

Good alphabets exhibit thermodynamical behavior enabling polynomial tiling algorithms

Bad alphabets show chaotic behavior resistant to polynomial solutions

The approach suggests a structured landscape within NP problems similar to phase transitions

Abstract

We introduce the idea that the P vs NP problem can have a finer structure. Given the NP complete problem of interest, the configurations space of the problem can be divided in (at least) two regions. In one region, polynomial algorithms to solve the NP complete problem of interest are available (and we discuss one possible realization inspire by the games of chess and go). In the second region the problem to find polynomial time algorithms is very similar to the problem to find polynomial time algorithms to determine the asymptotic behavior of discrete dynamical systems in the chaotic regime. We cannot exclude the existence of a third region which separates the first two: this region would have the characteristics of the edge of chaos. We focuss on the Wang tiling problem of an N X N square (with N large): here a Wang tiles set Gamma is an alphabet. We construct a statistical-physics inspired heuristic which allows to define good alphabets as the ones with a good thermodynamical behavior. For (a suitable subclass of) good alphabets we construct an algortihm which, in polynomial time, determines how to tile the N x N square. On the other hand, for bad alphabets, we observe a chaotic behavior. The Cook-Levin theorem advocate a similar pattern for all the NP-complete problems.