Simple heuristics often show a remarkable performance in practice for optimization problems. Worst-case analysis often falls short of explaining this performance. Because of this, "beyond worst-case analysis" of algorithms has recently gained a lot of attention, including probabilistic analysis of algorithms. The instances of many optimization problems are essentially a discrete metric space. Probabilistic analysis for such metric optimization problems has nevertheless mostly been conducted on instances drawn from Euclidean space, which provides a structure that is usually heavily exploited in the analysis. However, most instances from practice are not Euclidean. Little work has been done on metric instances drawn from other, more realistic, distributions. Some initial results have been obtained by (Algorithmica, 2013), who have used random shortest path metrics on complete graphs to analyze heuristics. The goal of this paper is to generalize these findings to non-complete graphs, especially Erdős-Renyi random graphs. A random shortest path metric is constructed by drawing independent random edge weights for each edge in the graph and setting the distance between every pair of vertices to the length of a shortest path between them with respect to the drawn weights. For such instances, we prove that the greedy heuristic for the minimum distance maximum matching problem, the nearest neighbor and insertion heuristics for the traveling salesman problem, and a trivial heuristic for the @math -median problem all achieve a constant expected approximation ratio. Additionally, we show a polynomial upper bound for the expected number of iterations of the 2-opt heuristic for the traveling salesman problem.
The Seventh Edition of the successful Introduction to Probability Models introduces elementary probability theory and stochastic processes. This book is particularly well-suited to those applying probability theory to the study of phenomena in engineering, management science, the physical and social sciences, and operations research. Skillfully organized, Introduction to Probability Models covers all essential topics. Sheldon Ross, a talented and prolific textbook author, distinguishes this book by his effort to develop in students an intuitive, and therefore lasting, grasp of probability theory. Ross' classic and best-selling text has been carefully and substantially revised. The Seventh Edition includes many new examples and exercises, with the majority of the new exercises being of the easier type. Also, the book introduces stochastic processes, stressing applications, in an easily understood manner. There is a comprehensive introduction to the applied models of probability that stresses intuition. Both professionals, researchers, and the interested reader will agree that this is the most solid and widely used book for probability theory. Features: * Provides a detailed coverage of the Markov Chain Monte Carlo methods and Markov Chain covertimes * Gives a thorough presentation of k-record values and the surprising Ignatov's * theorem * Includes examples relating to: "Random walks to circles," "The matching rounds problem," "The best prize problem," and many more * Contains a comprehensive appendix with the answers to approximately 100 exercises from throughout the text * Accompanied by a complete instructor's solutions manual with step-by-step solutions to all exercises New to this edition: * Includes many new and easier examples and exercises * Offers new material on utilizing probabilistic method in combinatorial optimization problems * Includes new material on suspended animation reliability models * Contains new material on random algorithms and cycles of random permutations Consider the minimal weights of paths between two points in a complete graph Kn with random weights on the edges, the weights being, for instance, uniformly distributed. It is shown that, asymptotically, this is log n n for two given points, that the maximum if one point is fixed and the other varies is 2 log n n, and that the maximum over all pairs of points is 3 log n n.Some further related results are given as well, including results on asymptotic distributions and moments, and on the number of edges in the minimal weight paths. Abstract We give explicit bounds for the tail probabilities for sums of independent geometric or exponential variables, possibly with different parameters. In 1957, Broadbent and Hammersley gave a mathematical formulation of percolation theory. Since then much work has been done in this field and has now led to first-passage percolation problems. In the following two examples we contrast the early formulation with its more recent developments. In this paper we explore first passage percolation (FPP) on the Erd˝ os-Renyi random graph Gn(pn), where each edge is given an independent exponential edge weight with rate 1. In the sparse regime, i.e., when npn → λ > 1, we find refined asymptotics both for the minimal weight of the path between uniformly chosen vertices in the giant component, as well as for the hopcount (i.e., the number of edges) on this minimal weight path. More precisely, we prove a central limit theorem for the hopcount, with asymptotic mean and variance both equal to λ (λ − 1) log n. Furthermore, we prove that the minimal weight centered by log n (λ − 1) converges in distribution. We also investigate the dense regime, where npn → ∞. We find that although the base graph is a ultra small (meaning that graph distances between uniformly chosen vertices are o(log n)), attaching random edge weights changes the geometry of the network completely. Indeed, the hopcount Hn satisfies the universality property that whatever be the value of pn, Hn log n → 1 in probability and, more precisely, (Hn−βn log n) √ log n, where βn = λn (λn− 1), has a limiting standard normal distribution. The constant βn can be replaced by 1 precisely when λn ≫ √ log n, a case that has appeared in the literature (under stronger conditions on λn) in (2, 12). We also find bounds for the maximal weight and maximal hopcount between vertices in the graph. This paper continues the investigation of FPP initiated in (2) and (3). Compared to the setting on the configuration model studied in (3), the proofs presented here are much simpler due to a direct relation between FPP on the Erd˝ os-Renyi random graph and thinned continuous-time branching processes. We consider the shortest paths between all pairs of nodes in a directed or undirected complete graph with edge lengths which are uniformly and independently distributed in [0, 1]. We show that die longest of these paths is bounded by c log n n almost surely, where c is a constant and n is the number of nodes. Our bound is the best possible up to a constant. We apply this result to some well-known problems and obtain several algorithmic improvements over existing results. Our results hold with obvious modifications to random (as opposed to complete) graphs and to any distribution of weights whose density is positive and bounded from below at a neighborhood of zero. As a corollary of our proof we get a new result concerning the diameter of random graphs.