Randomised techniques in combinatorial algorithmics
Probabilistic techniques are becoming more and more important in Computer Science. Some of them are useful for the analysis of algorithms. The aim of this thesis is to describe and develop applications of these techniques. We first look at the problem of generating a graph uniformly at random from the set of all unlabelled graphs with n vertices, by means of efficient parallel algorithms. Our model of parallel computation is the well-known parallel random access machine (PRAM). The algorithms presented here are among the first parallel algorithms for random generation of combinatorial structures. We present two different parallel algorithms for the uniform generation of unlabelled graphs. The algorithms run in O(log2 n) time with high probability on an EREW PRAM using O(n2) processors. Combinatorial and algorithmic notions of approximation are another important thread in this thesis. We look at possible ways of approximating the parameters that describe the phase transitional behaviour (similar in some sense to the transition in Physics between solid and liquid state) of two important computational problems: that of deciding whether a graph is colourable using only three colours so that no two adjacent vertices receive the same colour, and that of deciding whether a propositional boolean formula in conjunctive normal form with clauses containing at most three literals is satisfiable. A specific notion of maximal solution and, for the second problem, the use of a probabilistic model called the (young) coupon collector allows us to improve the best known results for these problems. Finally we look at two graph theoretic matching problems. We first study the computational complexity of these problems and the algorithmic approximability of the optimal solutions, in particular classes of graphs. We also derive an algorithm that solves one of them optimally in linear time when the input graph is a tree as well as a number of non-approximability results. Then we make some assumptions about the input distribution, we study the expected structure of these matchings and we derive improved approximation results on several models of random graphs.