Abstract

. In this paper we present the research that has been done with Linear Dynamical Systems to generate almost uniformly elements from a given set, and thus approximate some hard counting problems. We also indicate how non-linear systems can help to parallelize the computation. Finally we outline possible applications of linear systems to formalize heuristics. 1. Introduction Many problems involving counting solutions of combinatorial structures are well known to be difficult. Valiant defined the class #P of computationally equivalent counting problems ([Val79b]). For many problems in this class, their decision counterpart is in P . It is known that, unless the polynomial hierarchy collapses, P 6= #P . This fact implies that for any #P -complete problem, exact counting is apparently intractable ([Pap94]). The most notorious of these problems is to compute the permanent of a dense matrix. That problem turns out to be equivalent to counting the number of perfect matchings in a dense bipar...

Citations