Phase transition in the assignment problem for random matrices

TL;DR

This paper investigates a phase transition in the assignment problem that delineates an easy matching phase from a complex TSP phase, enhancing understanding of computational difficulty in NP problems.

Contribution

It provides an analytic and numerical analysis of the phase transition in the assignment problem, linking it to the complexity of related combinatorial problems.

Findings

Identifies a phase transition separating easy and hard problem instances.

Connects the transition to the complexity of the traveling salesman problem.

Offers insights into algorithmic challenges in NP problems.

Abstract

We report an analytic and numerical study of a phase transition in a P problem (the assignment problem) that separates two phases whose representatives are the simple matching problem (an easy P problem) and the traveling salesman problem (a NP-complete problem). Like other phase transitions found in combinatoric problems (K-satisfiability, number partitioning) this can help to understand the nature of the difficulties in solving NP problems an to find more accurate algorithms for them.