2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/59292We 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. 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. As an application, we present a new randomized polynomial time algorithm which approximates the permanent of a 0-1 matrix by solving a small number of Assignment problems.34 pagesCombinatoricsMetric Geometry05A16, 05C70, 46N10, 52C45, 60C05, 60D05, 68W20, 68R05text