In the area of statistical physics, Monte Carlo algorithms based on Markov chain simulation have been in use for many years. The validity of these algorithms depends crucially on the rate of convergence to equilibrium of the Markov chain being simulated. Unfortunately, the classical theory of stochastic processes hardly touches on the sort of non-asymptotic analysis required in this application. As a consequence, it had previously not been possible to make useful, mathematically rigorous statements about the quality of the estimates obtained. Within the last ten years, analytical tools have been devised with the aim of correcting this deficiency. As well as permitting the analysis of Monte Carlo algorithms for classical problems in statistical physics, the introduction of these tools has spurred the development of new approximation algorithms for a wider class of problems in combinatorial enumeration and optimization. The “Markov chain Monte Carlo ” method has been applied to a variety of such problems, and often provides the only known efficient (i.e., polynomial time) solution technique.

We present a unified framework for designing polynomial time approximation schemes (PTASs) for "dense" instances of many NP-hard optimization problems, including maximum cut, graph bisection, graph separation, minimum k-way cut with and without specified terminals, and maximum 3-satisfiability. By dense graphs we mean graphs with minimum degree Ω(n), although our algorithms solve most of these problems so long as the average degree is Ω(n). Denseness for non-graph problems is defined similarly. The unified framework begins with the idea of exhaustive sampling: picking a small random set of vertices, guessing where they go on the optimum solution, and then using their placement to determine the placement of everything else. The approach then develops into a PTAS for approximating certain smooth integer programs where the objective function and the constraints are "dense" polynomials of constant degree.

Problems such as bisection, graph coloring, and clique are generally believed hard in the worst case. However, they can be solved if the input data is drawn randomly from a distribution over graphs containing acceptable solutions. In this paper we show that a simple spectral algorithm can solve all three problems above in the average case, as well as a more general problem of partitioning graphs based on edge density. In nearly all cases our approach meets or exceeds previous parameters, while introducing substantial generality. We apply spectral techniques, using foremost the observation that in all of these problems, the expected adjacency matrix is a low rank matrix wherein the structure of the solution is evident.

The theory of random graphs has been mainly concerned with structural properties, in particular the most likely values of various graph invariants -- see Bollob`as [21]. There has been increasing interest in using random graphs as models for the average case analysis of graph algorithms. In this paper we survey some of the results in this area. 1 Introduction The theory of random graphs as initiated by Erdos and R'enyi [52] and developed along with others, has been mainly concerned with structural properties, in particular the most likely values of various graph invariantss -- see Bollob`as [21]. There has been increasing interest in using random graphs as models for the average case analysis of graph algorithms. We would like in this paper to survey some of the results in this area. We hope to be fairly comprehensive in terms of the areas we tackle and so depth will be sacrificed in favour of breadth. One attractive feature of average case analysis is that it banishes the pessimism o...

this paper, we give a rigorous analysis of such a "go with the winners" scheme in the concrete setting of searching for a deep leaf in a tree. There are two relevant parameters of the tree: its depth d, and another parameter which is a measure of the imbalance of the tree. We prove that the running time of the "go with the winners" scheme (to achieve 99% probability of success) is bounded by a polynomial in d and . By contrast, the simple restart scheme: run several independent

Abstract not found

We analyze the behavior of hill-climbing algorithms for the minimum bisection problem on instances drawn from the "planted bisection" random graph model, Gn;p;q , previously studied in [3, 4, 10, 12, 15, 9, 7]. This is one of the few problem distributions for which various popular heuristic methods, such as simulated annealing, have been proven to succeed. However, it has been open whether these sophisticated methods were necessary, or whether simpler heuristics would also work. Juels [15] made the first progress towards an answer by showing that simple hill-climbing does suffice for very wide separations between p and q.

We introduce a variant of Aldous and Vazirani's "Go with the winners" algorithm that can be used for search graphs that are not trees. We analyze the algorithm in terms of the properties of a tree-decomposition of the search graph. We show a large class of distributions for search graphs so that "Go with the winners" works well with high probability for almost all graphs from the distribution. We also give a sufficient combinatorial property that ensures good performance.

Abstract We discuss the problem of finding a satisfying assignment for a randomly chosen satisfiable 3- CNF with n variables and \Delta n clauses, where \Delta is much greater than the satisfiability threshold. We show that for \Delta = \Omega (log n), a trivial algorithm solves this problem. Our proof follows by showing that for large clause density, the distribution over random satisfiable 3-CNFs is statistically close to the planted-SAT distribution. This latter distribution is much easier to analyze, and it is for this distribution that we present our algorithm. Keywords: Satisfiability, 3-SAT, Random 3-CNF, planted SAT.

Let S(v) be a function defined on the vertices v of the infinite binary tree. One algorithm to seek large positive values of S is the Metropolis-type Markov chain (Xn ) defined by P (Xn+1 = wjXn = v) = 1 3 e b(S(w)\GammaS(v)) 1 + e b(S(w)\GammaS(v)) for each neighbor w of v, where b is a parameter ("1=temperature") which the user can choose. We introduce and motivate study of this algorithm under a probability model for the objective function S, in which S is "tree-indexed simple random walk", that is the increments ¸(e) = S(w) \Gamma S(v) along parent-child edges e = (v; w) are independent and P (¸ = 1) = p; P (¸ = \Gamma1) = 1 \Gamma p. This algorithm has a "speed" r(p; b) = lim n n \Gamma1 ES(Xn ). We study the speed via a mixture of rigorous arguments, non-rigorous arguments and Monte Carlo simulations, and compare with a deterministic greedy algorithm which permits rigorous analysis. Formalizing the non-rigorous arguments presents a challenging problem. Mathematically, th...