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We consider a refinement of the partition function of graph homomorphisms andpresent a quasi-polynomial algorithm to compute it in a certain domain. As acorollary, we obtain quasi-polynomial algorithms for computing partitionfunctions for independent sets, perfect matchings, Hamiltonian cycles and densesubgraphs in graphs as well as for graph colorings. This allows us to tellapart in quasi-polynomial time graphs that are sufficiently far from having astructure of a given type (i.e., independent set of a given size, Hamiltoniancycle, etc.) from graphs that have sufficiently many structures of that type,even when the probability to hit such a structure at random is exponentiallysmall.

Computing the partition function for graph homomorphisms with multiplicities