Stochastic models of optimization problems

Research Project, 2010 – 2012

We study optimization problems on graphs with random edge weights. In the simplest so-called mean field model, the graph is the complete graph on n vertices, and the edge weights are iid from exp(1)-distribution. Three classical optimization problems are minimum spanning tree, minimum matching, and the traveling salesman problem. In all three, the total cost of the optimum solution converges in probability as n tends to infinity, and the exact value of the limits can be found analytically. For the spanning tree it is zeta(3), and for minimum matching, pi^2/12. The limit for the TSP, approximately 2.04, was recently established by myself. There is a fascinating interplay between exact combinatorial methods, asymptotic probabilistic methods, and non-rigorous renormalization arguments originating in physics. The limit for minimum matching was conjectured by the physicists M. Mézard and G. Parisi in the 1980´s, and subsequently established by asymptotic methods by D. Aldous. There was also an exact conjecture for the corresponding problem on the bipartite graph, that the expected cost of the minimum perfect matching should be 1+1/4+...+1/n^2. This was established independently by two teams in 2003, and I have later generalized it in several directions. My current research in this area aims at making the renormalization methods rigorous. This can be seen as part of a much larger programme of putting the replica method of statistical physics on a firm mathematical ground.

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