Showing papers by "Yuri Rabinovich published in 2012"

On multiplicative λ-approximations and some geometric applications

Abstract: Let F be a set system over an underlying finite set X, and let μ be a nonnegative measure over X. I.e., for every S ⊆ X, μ(S) = ΣxeS μ(x). A measure μ* on X is called a multiplicative λ-approximation of μ on (F, X) if for every S e F it holds that aμ(S) ≤ μ* (S) ≤ bμ(S), and b/a = λ ≥ 1. The central question raised and partially answered in the present paper is about the existence of meaningful structural properties of F implying that for any μ on X there exists an 1+e/1-e-approximation μ* supported on a small subset of X.It turns out that the parameter that governs the support size of a multiplicative approximation is the triangular rank of F, trk(F). It is defined as the maximal length of a sequence of sets {Si}ti=1 in F such that for all 1

Constant Approximation Algorithms for Embedding Graph Metrics into Trees and Outerplanar Graphs

Abstract: In this paper, we present a simple factor 6 algorithm for approximating the optimal multiplicative distortion of embedding a graph metric into a tree metric (thus improving and simplifying the factor 100 and 27 algorithms of Bǎdoiu et al. (Proceedings of the eighteenth annual ACM–SIAM symposium on discrete algorithms (SODA 2007), pp. 512–521, 2007) and Bǎdoiu et al. (Proceedings of the 11th international workshop on approximation algorithms for combinatorial optimization problems (APPROX 2008), Springer, Berlin, pp. 21–34, 2008)). We also present a constant factor algorithm for approximating the optimal distortion of embedding a graph metric into an outerplanar metric. For this, we introduce a general notion of a metric relaxed minor and show that if G contains an α-metric relaxed H-minor, then the distortion of any embedding of G into any metric induced by a H-minor free graph is ≥α. Then, for H=K 2,3, we present an algorithm which either finds an α-relaxed minor, or produces an O(α)-embedding into an outerplanar metric.

Local Versus Global Properties of Metric Spaces

Abstract: Motivated by applications in combinatorial optimization, we study the extent to which the global properties of a metric space, and especially its embeddability into $\ell_1$ with low distortion, are determined by the properties of its small subspaces. We establish both upper and lower bounds on the distortion of embedding locally constrained metrics into various target spaces. Other aspects of locally constrained metrics are studied as well, in particular, how far are those metrics from general metrics.