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

Alejandro Mata Ali

TL;DR

The paper introduces MeLoCoToN, a general framework that turns any well-posed combinatorial problem into an explicit tensor-network equation whose solution is obtained by contracting a Tensor Logical Circuit (TLC). It builds problem-specific classical logical circuits (LSTC/LSVC/LSMC), tensorizes them via Circuit Tensorization and IOI indexing, and extracts variable values through half partial trace iterations, optionally enhanced by the Motion Onion methods. The framework encompasses a wide range of problem classes (QUBO, QUDO, CSP, TSP, knapsack, scheduling, etc.) and provides concrete tensor constructions for each, along with strategies to approximate or speed up contraction. While acknowledging exponential costs in many cases, the work argues that a polynomial-time contractible physical system would imply that all NP-hard problems are tractable, highlighting a new mathematical lens for combinatorial analysis and potential cross-pollination with quantum-inspired or heuristic approaches. The paper also outlines avenues for optimization, integration with genetic algorithms, and extensions to higher-order and multivariate problems, making MeLoCoToN a unifying, albeit theoretical, perspective on exact solution equations for combinatorial problems.

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.

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

TL;DR

The paper introduces MeLoCoToN, a general framework that turns any well-posed combinatorial problem into an explicit tensor-network equation whose solution is obtained by contracting a Tensor Logical Circuit (TLC). It builds problem-specific classical logical circuits (LSTC/LSVC/LSMC), tensorizes them via Circuit Tensorization and IOI indexing, and extracts variable values through half partial trace iterations, optionally enhanced by the Motion Onion methods. The framework encompasses a wide range of problem classes (QUBO, QUDO, CSP, TSP, knapsack, scheduling, etc.) and provides concrete tensor constructions for each, along with strategies to approximate or speed up contraction. While acknowledging exponential costs in many cases, the work argues that a polynomial-time contractible physical system would imply that all NP-hard problems are tractable, highlighting a new mathematical lens for combinatorial analysis and potential cross-pollination with quantum-inspired or heuristic approaches. The paper also outlines avenues for optimization, integration with genetic algorithms, and extensions to higher-order and multivariate problems, making MeLoCoToN a unifying, albeit theoretical, perspective on exact solution equations for combinatorial problems.

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.

Paper Structure

This paper contains 93 sections, 3 theorems, 165 equations, 45 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Definition of the general method MeLoCoToN
  3. Quantum computing explanation of the method
  4. Definition of the problem variables and rewriting of the functions
  5. Creation of the associated classical logical circuit
  6. Inversion Problem
  7. Sum of two numbers in binary
  8. Multiply two numbers in binary
  9. Constraint Satisfaction Problem
  10. Single One Input
  11. Optimization Problem
  12. Linear function
  13. Quadratic function with a single neighbor in a linear chain
  14. Natural sum total function problem
  15. Creation of the associated logical tensor network
  16. ...and 78 more sections

Key Result

Theorem 1

Given a combinatorial problem, be it an inversion problem, a constraint satisfaction problem or an optimization problem, there is an exact explicit equation for its solution (or solutions).

Figures (45)

  • Figure 1: General scheme of ideas presented.
  • Figure 2: LSTC to add two numbers $a$ and $b$ to obtain a number $c$.
  • Figure 3: LSTC to add two numbers $a$ and $b$ to obtain a number $c$ if $w=1$.
  • Figure 4: LSTC that performs the multiplication of two numbers $a$ and $b$.
  • Figure 5: LSVC to determine if a number has only one bit on $1$.
  • ...and 40 more figures

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3