Showing papers by "Yuri Rabinovich published in 1999"

Cuts, Trees and ℓ 1 -Embeddings of Graphs*

Abstract: Motivated by many recent algorithmic applications, this paper aims to promote a systematic study of the relationship between the topology of a graph and the metric distortion incurred when the graph is embedded into \math space. The main results are: 1. Explicit constant-distortion embeddings of all series-parallel graphs, and all graphs with bounded Euler number. These are thus the first natural families known to have constant distortion (strictly greater than 1). Using the above embeddings, we obtain algorithms to approximate the sparsest cut in such graphs to within a constant factor. 2. A constant-distortion embedding of outerplanar graphs into the restricted class of \math-metrics known as "dominating tree metrics". We also show a lower bound of \math on the distortion for embeddings of series-parallel graphs into (distributions over) dominating tree metrics. This shows, surprisingly, that such metrics approximate distances very poorly even for families of graphs with low treewidth, and excludes the possibility of using them to explore the finer structure of \math-embeddability.

Techniques for bounding the convergence rate of genetic algorithms

Abstract: The main purpose of the present paper is the study of computational aspects, and primarily the convergence rate, of genetic algorithms (GAs). Despite the fact that such algorithms are widely used in practice, little is known so far about their theoretical properties, and in particular about their long-term behavior. This situation is perhaps not too surprising, given the inherent hardness of analyzing nonlinear dynamical systems, and the complexity of the problems to which GAs are usually applied. In the present paper we concentrate on a number of very simple and natural systems of this sort, and show that at least for these systems the analysis can be properly carried out. Various properties and tight quantitative bounds on the long-term behavior of such systems are established. It is our hope that the techniques developed for analyzing these simple systems prove to be applicable to a wider range of genetic algorithms, and contribute to the development of the mathematical foundations of this promising optimization method. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 14, 111–138, 1999

Cuts, trees and l/sub 1/-embeddings of graphs

Abstract: Motivated by many recent algorithmic applications, the paper aims to promote a systematic study of the relationship between the topology of a graph and the metric distortion incurred where the graph is embedded into l/sub 1/ space. The main results are: 1. Explicit constant-distortion embeddings of all series parallel graphs, and all graphs with bounded Euler number. These are thus the first natural families known to have constant distortion (strictly greater than 1). Using the above embeddings, we obtain algorithms to approximate the sparsest cut in such graphs to within a constant factor. 2) A constant-distortion embedding of outerplanar graphs into the restricted class of l/sub 1/-metrics known as "dominating tree metrics". We also show a lower bound of /spl Omega/(log n) on the distortion for embeddings of series-parallel graphs into (distributions over) dominating tree metrics. This shows, surprisingly, that such metrics approximate distances very poorly even for families of graphs with low tree width, and excludes the possibility of using them to explore the finer structure of l/sub 1/-embeddability.

Intersection Properties of Families of Convex (n,d)-Bodies

Abstract: We study a multicomponent generalization of Helly's theorem. An (n,d)-body K is an ordered n -tuple of d -dimensional sets, K= < K 1 , . . . ,K n > . A family $\cal F$ of (n,d)-bodiesis weakly intersecting if there exists an n -point p = < p 1 , . . . , p n > such that for every $K \in {\cal F}$ there exists an index 1 $\leq i \leq n$ for which p i ∈ K i . A family $\cal F$ of (n,d)-bodies is strongly intersecting if there exists an index i such that $\bigcap_{K\in{\cal F}} K_i eq \emptyset$ . The main question addressed in this paper is: What is the smallest number H(n,d), such that for every finite family of convex (n,d)-bodies, if every H(n,d) of them are strongly intersecting, then the entire family is weakly intersecting? We establish some basic facts about H(n,d) , and also prove an upper bound $H(n,d)\leq (\lfloor \log_2 (n+1) \rfloor + 1)^d$ . In addition, we introduce and discuss two interesting related questions of a combinatorial-topological nature.

Optimal Bounds on Tail Probabilities: A Study of an Approach

Abstract: In Computer Science and Statistics it is often desirable to obtain tight bounds on the decay rate of probabilies of the type \(\Pr \left\{ {{S_n} - E\left[ {{S_n}} \right] \geqslant na} \right\},\), where S n is a sum of independent random variables \(\left\{ {{X_i}} \right\}_{i = 1}^n\). This is usually done by means of Chernoff inequality, or the more general Hoeffding inequality. The latter inequality is assymptotically optimal as far as the expectations of X i -s go, but ceases to be so when the variances are also given. The variances are taken into account in the stronger Bennett inequality, which despite its potential usefulness is virtually unknown in CS community.