Phase transition in the assignment problem for random matrices

– 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. In the theory of computational complexity, two paradigmatic problems representative of the classes NP -complete and P are the traveling salesman and the assignment problem. Traveling salesman problem (TSP) requires N points or cities, with distances di,j between them, for which we must find the closed tour of minimum length that visits each city once. In the assignment or bipartite matching problem (AP), we have N objects of two classes (say A and B) with distances di,j between each object i of the class A and the object j of the class B. We must assign at each object i of the class A one and only one object σ(i) of the class B in such a way that the total distance ∑N i=1 di,σ(i) is a minimum, so we can write D(AP) as: D(AP ) = Min(σ∈SN ) [ N