Robert Krauthgamer

Bounded geometries, fractals, and low-distortion embeddings

The doubling constant of a metric space (X, d) is the smallest value /spl lambda/ such that every ball in X can be covered by /spl lambda/ balls of half the radius. The doubling dimension of X is then defined as dim (X) = log/sub 2//spl lambda/. A metric (or sequence of metrics) is called doubling precisely when its doubling dimension is bounded. This is a robust class of metric spaces which contains many families of metrics that occur in applied settings. We give tight bounds for embedding doubling metrics into (low-dimensional) normed spaces. We consider both general doubling metrics, as well as more restricted families such as those arising from trees, from graphs excluding a fixed minor, and from snowflaked metrics. Our techniques include decomposition theorems for doubling metrics, and an analysis of a fractal in the plane according to T. J. Laakso (2002). Finally, we discuss some applications and point out a central open question regarding dimensionality reduction in L/sub 2/.

On the hardness of approximating MULTICUT and SPARSEST-CUT

We show that the MULTICUT, SPARSEST-CUT, and MIN-2CNF/spl equiv/DELETION problems are NP-hard to approximate within every constant factor, assuming the unique games conjecture of Khot [STOC, 2002]. A quantitatively stronger version of the conjecture implies inapproximability factor of /spl Omega/(log log n).

Polylogarithmic inapproximability

We provide the first hardness result of a polylogarithmic approximation ratio for a natural NP-hard optimization problem. We show that for every fixed ε>0, the GROUP-STEINER-TREE problem admits no efficient log2-ε k approximation, where k denotes the number of groups (or, alternatively, the input size), unless NP has quasi polynomial Las-Vegas algorithms. This hardness result holds even for input graphs which are Hierarchically Well-Separated Trees, introduced by Bartal [FOCS, 1996]. For these trees (and also for general trees), our bound is nearly tight with the log-squared approximation currently known. Our results imply that for every fixed ε>0, the DIRECTED-STEINER TREE problem admits no log2-ε n--approximation, where n is the number of vertices in the graph, under the same complexity assumption.

Hardness of Approximation for Vertex-Connectivity Network Design Problems

In the survivable network design problem (SNDP), the goal is to find a minimum-cost spanning subgraph satisfying certain connectivity requirements. We study the vertex-connectivity variant of SNDP in which the input specifies, for each pair of vertices, a required number of vertex-disjoint paths connecting them. We give the first strong lower bound on the approximability of SNDP, showing that the problem admits no efficient

Improved lower bounds for embeddings into L 1

2^{\log^{1-\epsilon} n}

How Hard Is It to Approximate the Best Nash Equilibrium

ratio approximation for any fixed

Algorithms on negatively curved spaces

\epsilon\! >\! 0

The Probable Value of the Lovász-Schrijver Relaxations for Maximum Independent Set

, unless

Fault-tolerant spanners: better and simpler

\NP\subseteq \DTIME(n^{\polylog(n)})

Asymmetric k -center is log * n -hard to approximate

. We show hardness of approximation results for some important special cases of SNDP, and we exhibit the first lower bound on the approximability of the related classical NP-hard problem of augmenting the connectivity of a graph using edges from a given set.