Explicit Solution Equation for Every Combinatorial Problem via Tensor Networks: MeLoCoToN

TL;DR

This paper introduces a universal method to derive explicit equations for any combinatorial problem using tensor networks, offering a new perspective for mathematical analysis despite computational limitations.

Contribution

It presents a novel approach to formulate combinatorial problems as explicit tensor network equations, bridging physics and computer science.

Findings

Every combinatorial problem has an exact tensor network equation.

Tensor network equations can be approximated by Matrix Product State compression.

If physical systems can efficiently contract these tensor networks, NP-Hard problems could be solved in polynomial time.

Abstract

In this paper we show that every combinatorial problem has an exact explicit equation that returns its solution. We present a method to obtain an equation that solves exactly any combinatorial problem, both inversion, constraint satisfaction and optimization, by obtaining its equivalent tensor network. This formulation only requires a basic knowledge of classical logical operators, at a first year level of any computer science degree. These equations are not necessarily computable in a reasonable time, nor do they allow to surpass the state of the art in computational complexity, but they allow to have a new perspective for the mathematical analysis of these problems. These equations computation can be approximated by different methods such as Matrix Product State compression. We also present the equations for numerous combinatorial problems. This work proves that, if there is a physical system capable of contracting in polynomial time the tensor networks presented, every NP-Hard problem can be solved in polynomial time.