Advanced Approximation Algorithms ( CMU 15 - 854 B , Spring 2008 ) Lecture 20 : Embeddings into Trees and L 1 Embeddings March 27 2008

where c(E(S, S̄)) is the sum of the weights of the edges that cross the cut, and D(S, S̄) is the sum of the demands of the pairs (si, ti) that are separated by the cut. We recall that optimizing over the set of cuts is equivalent to optimizing over `1 metrics, and is NP-hard. Instead, we may optimize over the set of all metrics. In this lecture, we bound the gap introduced by this relaxation by showing how these metrics embed into `1 with low distortion. From last time, we have a theorem of Bourgain: Theorem 1.1 (Bourgain 85 [1]). Any n point metric d admits an α-distortion embedding into `p for any 1 ≤ p ≤ ∞ with α = O(log n). This embedding is into 2 dimensions, however! Fortunately, Linial, London, and Rabinovich [3] proved that it is possible to get such an embedding into online O(log n) dimensions. In this lecture, we will show a somewhat weaker result: We will achieve an α = O(log n) embedding into `1 using poly(n, dmax/dmin) dimensions, where dmax is the maximum distance between any two points according to metric d, and dmin is the minimum distance. We note that dmax/dmin could potentially be exponentially large – but we won’t worry about this detail.