Random Generation and Approximate Counting of Combinatorial Structures

TL;DR

This thesis explores classes of NP relations where random generation and approximate counting are efficiently solvable, introducing ambiguous descriptions and analyzing the complexity limits of derandomizing heuristics in combinatorial optimization.

Contribution

It introduces the concept of ambiguous descriptions for random generation and approximate counting, and analyzes the complexity of derandomizing heuristics for combinatorial optimization.

Findings

Efficient random generation is possible even when ranking is infeasible.

Ambiguous descriptions are effective for formal language applications.

Derandomizing certain heuristics can be #P-hard.

Abstract

The aim of this thesis is to determine classes of NP relations for which random generation and approximate counting problems admit an efficient solution. Since efficient rank implies efficient random generation, we first investigate some classes of NP relations admitting efficient ranking. On the other hand, there are situations in which efficient random generation is possible even when ranking is computationally infeasible. We introduce the notion of ambiguous description as a tool for random generation and approximate counting in such cases and show, in particular, some applications to the case of formal languages. Finally, we discuss a limit of an heuristic for combinatorial optimization problems based on the random initialization of local search algorithms showing that derandomizing such heuristic can be, in some cases, #P-hard.