# A Hypergraph Based Approach for the 4-Constraint Satisfaction Problem Tractability

## TL;DR

This paper introduces a hypergraph-based method for efficiently solving a specific subclass of constraint satisfaction problems called 4-CSP, demonstrating polynomial time algorithms for their resolution.

## Contribution

It provides the first graph-based proofs of 4-CSP tractability and develops polynomial algorithms based on hypergraph closure and positive linear dependence theory.

## Key findings

- First graph-based proofs of 4-CSP tractability.
- Polynomial time algorithms for 4-CSP resolution.
- Algorithms based on hypergraph closure and linear dependence.

## Abstract

Constraint Satisfaction Problem (CSP) is a framework for modeling and solving a variety of real-world problems. Once the problem is expressed as a finite set of constraints, the goal is to find the variables' values satisfying them. Even though the problem is in general NP-complete, there are some approximation and practical techniques to tackle its intractability. One of the most widely used techniques is the Constraint Propagation. It consists in explicitly excluding values or combination of values for some variables whenever they make a given subset of constraints unsatisfied. In this paper, we deal with a CSP subclass which we call 4-CSP and whose constraint network infers relations of the form: $\{ x \sim \alpha, x-y \sim \beta , (x-y) - (z-t) \sim \lambda \}$, where $x, y, z$ and $t$ are real variables, $\alpha , \beta$ and $ \lambda $ are real constants and $ \sim \in \{\leq , \geq \} $. The paper provides the first graph-based proofs of the 4-CSP tractability and elaborates algorithms for 4-CSP resolution based on the positive linear dependence theory, the hypergraph closure and the constraint propagation technique. Time and space complexities of the resolution algorithms are proved to be polynomial.