Combinatorial decomposition approaches for efficient counting and random generation FPTASes

TL;DR

This paper introduces a novel floating-point approximation method to develop efficient FPTASes for counting and sampling problems in combinatorics, including DAGs and knapsack solutions, improving upon existing algorithms in speed and simplicity.

Contribution

It presents the first FPTASes with relative error for counting and generating DAGs and extends the approach to various combinatorial counting problems, simplifying and speeding up existing algorithms.

Findings

FPTAS with 1 ± ε relative error for counting and generating DAGs.

Improved FPTAS for 0/1 Knapsack counting problem.

Faster and simpler FPTAS for counting solutions in weighted DAGs.

Abstract

Given a combinatorial decomposition for a counting problem, we resort to the simple scheme of approximating large numbers by floating-point representations in order to obtain efficient Fully Polynomial Time Approximation Schemes (FPTASes) for it. The number of bits employed for the exponent and the mantissa will depend on the error parameter $0 < \varepsilon \leq 1$ and on the characteristics of the problem. Accordingly, we propose the first FPTASes with $1 \pm \varepsilon$ relative error for counting and generating uniformly at random a labeled DAG with a given number of vertices. This is accomplished starting from a classical recurrence 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 approach works also with problems where we are given a compact representation of a combinatorial ensemble and we are asked to count and sample elements from it. We employ here the floating-point approximation method to transform the classic pseudo-polynomial algorithm for counting 0/1 Knapsack solutions into a very simple FPTAS with $1 - \varepsilon$ relative error. Its complexity improves upon the recent result (\v{S}tefankovi\v{c} et al., SIAM J. Comput., 2012), and, when $\varepsilon^{-1} = \Omega(n)$, also upon the best-known randomized algorithm (Dyer, STOC, 2003). To show the versatility of this technique, we also apply it to a recent generalization of the problem of counting 0/1 Knapsack solutions in an arc-weighted DAG, obtaining a faster and simpler FPTAS than the existing one.