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Approximating extent measures of points
TLDR
We present a general ϵ-approximation algorithm for approximating various descriptors of the extent of a set <i>P</i> of points in R<sup><i>d</i></sup> with the property that (1 − ϵ)μ(<i> P</i>) ≤ μ(< i>Q</i>. Expand
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Geometric Approximation via Coresets
The paradigm of coresets has recently emerged as a powerful tool for efficiently approximating various extent measures of a point set P . Using this paradigm, one quickly computes a small subset Q ofExpand
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The Planar k-Means Problem is NP-Hard
TLDR
We show that the k-means problem is NP-hard even for instances in the plane, answering an open question posed by Dasgupta [6]. Expand
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Improved Approximation Algorithms for Geometric Set Cover
TLDR
We show that polynomial-time approximation algorithms with provable performance exist, under a certain general condition: that for a random subset $R\subset S$ and nondecreasing function f(·), there is a decomposition of the complement ${\Bbb U}\backslash\bigcup (R)$ into an expected at most f(|R|) regions, each region of a particular simple form. Expand
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Weighted geometric set cover via quasi-uniform sampling
TLDR
We employ the idea of quasi-uniform sampling to obtain improved approximation guarantees in the weighted setting for a large class of problems for which such guarantees were known in the unweighted case. Expand
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Efficient Algorithms for Approximating Polygonal Chains
Abstract. We consider the problem of approximating a polygonal chain C by another polygonal chain C' whose vertices are constrained to be a subset of the set of vertices of C . The goal is toExpand
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Market equilibrium via the excess demand function
TLDR
We consider the problem of computing market equilibria and show three results. (i) For exchange economies satisfying weak gross substitutability we analyze a simple discrete version of tâtonnement, and prove that it converges to an approximate equilibrium in polynomial time. (ii) For Fisher's model, we extend the frontier of tractability by developing an algorithm that applies well beyond the homothetic case and the gross substitutes case. Expand
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On the polynomial time computation of equilibria for certain exchange economies
TLDR
We obtain alternative polynomial-time algorithms for computing equilibria with linear, Cobb-Douglas, a range of CES, as well as certain other non-homogeneous utility functions that satisfy weak gross substitutability. Expand
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Buffer minimization using max-coloring
Given a graph <i>G</i> = (<i>V,E</i>) and positive integral vertex weights <i>w</i> : <i>V</i> → N, the <i>max-coloring problem</i> seeks to find a proper vertex coloring of <i>G</i> whose colorExpand
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Leontief economies encode nonzero sum two-player games
TLDR
We give a reduction from two-player games to a special family of Leontief exchange economies, which are guaranteed to have equilibria, with the property that the Nash Equilibria of any game are in one-to-one correspondence with the equilibrium of the corresponding economy. Expand
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