It is shown that the minimum cut ratio is within a factor of O(log k) of the maximum concurrent flow for k-commodity flow instances with arbitrary capacities and demands. This improves upon the previously best known bound of O(log 2 k) [27], and is existentially tight, up to a constant factor. An algorithm for finding a cut with ratio within a factor of O(log k) of the maximum concurrent flow, and thus of the optimal min cut ratio, is presented.
We investigate variants of Lloyd's heuristic for clustering high dimensional data in an attempt to explain its popularity (a half century after its introduction) among practitioners, and in order to suggest improvements in its application. We propose and justify a clusterability criterion for data sets. We present variants of Lloyd's heuristic that quickly lead to provably near-optimal clustering solutions when applied to well-clusterable instances. This is the first performance guarantee for a variant of Lloyd's heuristic. The provision of a guarantee on output quality does not come at the expense of speed: some of our algorithms are candidates for being faster in practice than currently used variants of Lloyd's method. In addition, our other algorithms are faster on well-clusterable instances than recently proposed approximation algorithms, while maintaining similar guarantees on clustering quality. Our main algorithmic contribution is a novel probabilistic seeding process for the starting configuration of a Lloyd-type iteration.
Given an undirected graph wit.h edge co&s and a subset of k nodes called terminals, a multiway cut is a subset of edges whose removal disconnects each terminal from the rest. ~iULTIW.~y CUT is the problem of finding a multiway cut of minimum cost.. Previously, a very simple combinatorial algorithm due to Dahlhaus, Johnson, Papadimitriou, Seymour, and %nnr-lkakis gave a performance guarantee of 2 (1 -$), In this paper, we present a new linear programming rslax&ion for ~fULTIW&Y CUT and a new approximation dgorithm based on it. The algorithm breaks the threshold of 2 for approximating MULTIWAY CUT, achieving a performance ratio of at. most 1.5 -$. This improves the previous result for every value of k. In particular, for k = 3 we get a ratio ofZ
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