A Probabilistic Attack on NP-complete Problems

TL;DR

This paper explores a probabilistic approach linking NP-complete problems to harmonic cost functions, suggesting continuous mathematics could offer new insights into longstanding complexity theory questions.

Contribution

It introduces a novel probabilistic framework connecting NP-complete problems to harmonic functions and their geometric properties.

Findings

Establishes an equivalence between NP-complete problems and harmonic cost function level sets.

Proposes that continuous mathematical methods may shed light on open problems in complexity theory.

Highlights the potential of harmonic analysis in understanding computational complexity.

Abstract

Using the probability theory-based approach, this paper reveals the equivalence of an arbitrary NP-complete problem to a problem of checking whether a level set of a specifically constructed harmonic cost function (with all diagonal entries of its Hessian matrix equal to zero) intersects with a unit hypercube in many-dimensional Euclidean space. This connection suggests the possibility that methods of continuous mathematics can provide crucial insights into the most intriguing open questions in modern complexity theory.