Detecting the finer structure of the P vs NP problem with statistical mechanics: the case of the Wang tiling problem
Fabrizio Canfora, Marco Cedeno
TL;DR
The paper argues that the P vs NP problem may exhibit a finer structure when viewed through the lens of Wang tilings, by partitioning alphabets into a Tractable Regime with polynomial-time tiling and a Chaotic Regime with intrinsic hardness, potentially separated by an Edge of Chaos. It introduces a statistical mechanics–inspired framework using observables like $S_{\Gamma}(n)=\log W_{\Gamma}(n)$ and a partition function $Z_{\Gamma}(\beta)$ to classify alphabets as good or bad, and demonstrates that for a subset of good alphabets a two-phase, Chess/Go–inspired algorithm yields polynomial-time tilings after an initial seed stage. The procedure relies on Brute-force seed generation up to $n_{\max}$ ( Phase I ) followed by a dressing/propagation approach from $n$ to $n+1$ ( Phase II ), enabling large-scale tilings for several alphabets (G1–G5) with varying levels of performance. The findings suggest a non-homogeneous tiling space and motivate the notion of a boundary region—possibly corresponding to an edge of chaos—impinging on the universality of NP-hardness, with implications for a refined understanding of the P vs NP landscape and future formalization of the conjectured Regime distinctions.
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.