Combinatorial decomposition approaches for efficient counting and random generation FPTASes

Given a combinatorial decomposition for a counting problem, we resort to the simplescheme of approximating large numbers by floating-point representations in order to obtainefficient Fully Polynomial Time Approximation Schemes (FPTASes) for it. The numberof bits employed for the exponent and the mantissa will depend on the error parameter0 < ε ≤ 1 and on the characteristics of the problem. Accordingly, we propose the firstFPTASes with 1 ± ε relative error for counting and generating uniformly at random alabeled DAG with a given number of vertices. This is accomplished starting from a classicalrecurrence for counting DAGs, whose values we approximate by floating-point numbers.After extending these results to other families of DAGs, we show how the same approachworks also with problems where we are given a compact representation of a combinatorialensemble and we are asked to count and sample elements from it. We employ here thefloating-point approximation method to transform the classic pseudo-polynomial algorithmfor counting 0/1 Knapsack solutions into a very simple FPTAS with 1− ε relative error. Itscomplexity improves upon the recent result (Štefankovič et al., SIAM J. Comput., 2012),and, when ε = Ω(n), also upon the best-known randomized algorithm (Dyer, STOC,2003). To show the versatility of this technique, we also apply it to a recent generalizationof the problem of counting 0/1 Knapsack solutions in an arc-weighted DAG, obtaining afaster and simpler FPTAS than the existing one.