2 research outputs found
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.Comment: 7 pages, 5 figures; accepted for publication in Europhys. Lett.
http://www.edpsciences.org/journal/index.cfm?edpsname=ep
On the number of -cycles in the assignment problem for random matrices
We continue the study of the assignment problem for a random cost matrix. We
analyse the number of -cycles for the solution and their dependence on the
symmetry of the random matrix. We observe that for a symmetric matrix one and
two-cycles are dominant in the optimal solution. In the antisymmetric case the
situation is the opposite and the one and two-cycles are suppressed. We solve
the model for a pure random matrix (without correlations between its entries)
and give analytic arguments to explain the numerical results in the symmetric
and antisymmetric case. We show that the results can be explained to great
accuracy by a simple ansatz that connects the expected number of -cycles to
that of one and two cycles.Comment: To appear in Journal of Statistical Mechanic