Optimal Cooperation and Submodularity for Computing Potts' Partition Functions with a Large Number of State

The partition function of the q-state Potts model with random ferromagnetic couplings in the large-q limit is generally dominated by the contribution of a single diagram of the high temperature expansion. Computing this dominant diagram amounts to minimizing a particular submodular function. We provide a combinatorial optimization algorithm, the optimal cooperation algorithm, which works in polynomial time for any lattice. Practical implementation and the speed of the method is also discussed.