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

2025-02-09

AI Summary

Constructing exact, closed-form analytical solution equations for arbitrary combinatorial problems—including inverse problems, constraint satisfaction, and optimization—remains fundamentally challenging. Method: We propose a tensor-network-based universal modeling framework: combinatorial logic is encoded as tensors, and solution equations are derived end-to-end via matrix product state (MPS) compression and symbolic tensor network contraction. Contribution/Results: We provide the first rigorous proof that every combinatorial problem admits an explicit solution equation; establish a constructive paradigm linking combinatorial structure to tensor networks; and identify a physical decidability criterion for NP-hardness—namely, if the associated tensor network admits exact contraction in polynomial time, then P = NP. The approach relies solely on undergraduate-level logic and linear algebra, and has successfully yielded closed-form solutions for multiple classical combinatorial problems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Matrix & Tensor Methods

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Exact explicit equation for combinatorial problems
• Tensor network equivalence for solving problems
• Potential polynomial-time solution for NP-Hard problems

Innovation

• Tensor networks solve combinatorial problems
• Exact equations via logical operators
• Matrix Product State compression approximation