The paper studies effective approximate solutions to combinatorial counting and uniform generation problems. Using a technique based on the simulation of ergodic Markov chains, it is shown that, for self-reducible structures, almost uniform generation is possible in polynomial time provided only that randomised approximate counting to within some arbitrary polynomial factor is possible in polyn...

Counting classes consist of languages deened in terms of the number of accepting computations of nondeterministic polynomial-time Turing machines. Well known examples of counting classes are NP, co-NP, P, and PP. Every counting class is a subset of P #PP1] , the class of languages computable in polynomial time using a single call to an oracle capable of determining the number of accepting paths...

The relationship between counting functions and logical expressibility is explored. The most well studied class of counting functions is #P, which consists of the functions counting the accepting computation paths of a nondeterministic polynomial-time Turing machine. For a logic L, #L is the class of functions on nite structures counting the tuples (T ; c) satisfying a given formula (T ; c) in ...

We present a randomized algorithm, which, given positive integers n and t and a real number 0 < < 1, computes the number |Σ(n, t)| of n× n non-negative integer matrices (magic squares) with the row and column sums equal to t within relative error . The computational complexity of the algorithm is polynomial in −1 and quasi-polynomial in N = nt, that is, of the order N log N . A simplified versi...

Counting and uniform sampling of directed acyclic graphs (DAGs) from a Markov equivalence class are fundamental tasks in graphical causal analysis. In this paper, we show that these can be performed polynomial time, solving long-standing open problem area. Our algorithms effective easily implementable. Experimental results the significantly outperform state-of-the-art methods.

We define a family of generalizations of the two-variable quandle polynomial. These polynomial invariants generalize in a natural way to eight-variable polynomial invariants of finite biquandles. We use these polynomials to define a family of link invariants which further generalize the quandle counting invariant.