Approximating a Finite Metric by a Small Number of Tree Metrics

Bartal [4, 5] gave a randomized polynomial time algorithm that given any n point metric G, constructs a tree T such that the expected stretch (distortion) of any edge is at most O(logn log logn). His result has found several applications and in particular has resulted in approximation algorithms for many graph optimization problems. However approximation algorithms based on his result are inherently randomized. In this paper we derandomize the use of Bartal’s algorithm in the design of approximation algorithms. We give an efficient polynomial time algorithm that given a finite n point metric G, constructs O(n logn) trees and a probability distribution on them such that the expected stretch of any edge of G in a tree chosen according to is at most O(logn log logn). Our result establishes that finite metrics can be probabilistically approximated by a small number of tree metrics. We obtain the first deterministic approximation algorithms for buy-at-bulk network design [2] and vehicle routing [7]; in addition we subsume results from our earlier work [8] on derandomization. Our main result is obtained by a novel view of probabilistic approximation of metric spaces as a deterministic optimization problem via linear programming. This view also provides a new proof of the result in [5] that might be easier to generalize. Supported by the Pierre and Christine Lamond Fellowship, an ARO MURI Grant DAAH04-96-1-0007 and NSF Award CCR-9357849, with matching funds from IBM, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. Email: [email protected]. ySupported by an IBM Cooperative Fellowship, an ARO MURI Grant DAAH04-96-1-0007 and NSF Award CCR-9357849, with matching funds from IBM, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. Email: [email protected]. zSupported by Army contracts DAAG55-98-1-0170/DAAG55-97-1-0221 and ONR N00014-98-10189. Email: [email protected]. xSupported by an ARO MURI Grant DAAH04-96-1-0007 and NSF Award CCR-9357849, with matching funds from IBM, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. Email: [email protected]. {Supported by Army contracts DAAG55-98-1-0170/DAAG55-97-1-0221 and ONR N00014-98-10189. Email: [email protected] We also show that graphs induced by points in <dp (ddimensional real normed space equipped with the lp norm) can be O(f(d; p) logn)-probabilistically approximated by tree metrics where f(d; p) = d1=p for 1 p 2 and f(d; p) = d1 1=p for 2 p. We use an improved graph partitioning algorithm for normed spaces that obliviously partitions the space into clusters of diameter at mostD such that the probability of two points u and v falling in different clusters is at most O(f(d; p) ku vkp=D). We also show that our clustering is optimal for all p by giving matching