Random Generation and Approximate Counting of Combinatorial Structures

Ad Ilenia, la persona che più di tutte ha cambiato la mia vita. INTRODUCTION Combinatorial counting problems have a long and distinguished history. Apart from their intrinsic interest, they arise naturally from investigations in numerous branches of mathematics and natural sciences and have given rise to a rich and beautiful theory. Ranking problems, which consist in determining the position of a given element in a well-ordered set, are closely related to counting. Random generation problems are less well studied but have a large number of computational applications. From the structural complexity viewpoint, the study of counting problems was initiated by Valiant (1979). A parallel approach to random generation problems was proposed by Jerrum, Valiant, and Vazirani (1986); in particular, they show how the standard reduction from generation to exact counting can be modified to yield an almost uniform generator giving only approximate counting estimates. They also locate the almost uniform generation and approximate counting problems for general NP relations 1 within the second level of the (probabilistic) polynomial time hierarchy (Stockmeyer, 1977). Finally, ranking has been studied by Huynh (1990) and by Goldberg and Sipser (1991) that considered it as a special kind of optimal compression. 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. More details follow. Ranking. We extend some results about ranking for formal languages to the case of NP relations , a fact that allows us to introduce two new classes of relations admitting efficient (i.e., polynomial time) random uniform generation. In particular, we prove that the classes of NP relations accepted by (i) unambiguous auxiliary pushdown automata working in polynomial time, and 1 by NP relations here we mean subsets R ⊆ Σ * × Σ * such that, …