Fully polynomial time approximation schemes (FPTAS) for some counting problems

TL;DR

This thesis presents new fully polynomial time approximation schemes for counting problems including m-tuples, contingency tables, and 0/1 knapsack, improving efficiency and providing the first FPTAS for some of these problems.

Contribution

It introduces the first FPTASs for counting m-tuples and significantly improves existing algorithms for contingency tables, using advanced approximation methods.

Findings

First FPTASs for counting m-tuples.

Significant improvement in contingency table counting algorithms.

Simple strongly polynomial algorithm for 0/1 knapsack counting.

Abstract

In this thesis we develop FPTASs for the counting problems of m-tuples, contingency tables with two rows, and 0/1 knapsack. For the problem of counting m-tuples, we design two algorithms, one is strongly polynomial. As far as we know, these are the first FPTASs for this problem. For the problem of counting contingency tables we improve significantly over the running time of existing algorithms. For the problem of counting 0/1 knapsack solutions, we design a simple strongly polynomial algorithm, with similar running times to the existing algorithms. Our results are derived by using, as well as expanding, the method of K-approximation sets and functions.