We develop general methods to obtain fast (polynomial time) estimates of the cardinality of a combinatorially defined set via solving some randomly generated optimization problems on the set. Examples include enumeration of perfect matchings in a graph, linearly independent subsets of a set of vectors and colored spanning subgraphs of a graph. Geometrically, we estimate the cardinality of a subset of the Boolean cube via the average distance from a point in the cube to the subset with respect to some distance function. We derive asymptotically sharp cardinality bounds in the case of the Hamming distance and show that for small subsets a suitably defined "randomized" Hamming distance allows one to get tighter estimates of the cardinality.
Bibliographical noteFunding Information:
The research of the first author was partially supported by NSF Grant DMS 9734138. The research of the second author was partially supported by a State of New Jersey grant.