Kasturi R. Varadarajan

Approximating extent measures of points

We present a general technique for approximating various descriptors of the extent of a set <i>P</i> of <i>n</i> points in R^<i>d</i> when the dimension <i>d</i> is an arbitrary fixed constant. For a given extent measure μ and a parameter ϵ > 0, it computes in time <i>O</i>(<i>n</i> + 1/ϵ^<i>O</i>(1)) a subset <i>Q</i> ⊆ <i>P</i> of size 1/ϵ^<i>O</i>(1), with the property that (1 − ϵ)μ(<i>P</i>) ≤ μ(<i>Q</i>) ≤ μ(<i>P</i>). The specific applications of our technique include ϵ-approximation algorithms for (i) computing diameter, width, and smallest bounding box, ball, and cylinder of <i>P</i>, (ii) maintaining all the previous measures for a set of moving points, and (iii) fitting spheres and cylinders through a point set <i>P</i>. Our algorithms are considerably simpler, and faster in many cases, than previously known algorithms.

The Planar k-Means Problem is NP-Hard

In the k-means problem, we are given a finite set S of points in

Improved Approximation Algorithms for Geometric Set Cover

\Re^m

Improved approximation algorithms for geometric set cover

, and integer k ≥ 1, and we want to find k points (centers) so as to minimize the sum of the square of the Euclidean distance of each point in S to its nearest center. We show that this well-known problem is NP-hard even for instances in the plane, answering an open question posed by Dasgupta [6].

Leontief economies encode nonzero sum two-player games

Given a collection S of subsets of some set

The planar k-means problem is NP-hard

{\Bbb U},

Weighted geometric set cover via quasi-uniform sampling

and

Efficient Computation of Equilibrium Prices for Markets with Leontief Utilities

{\Bbb M}\subset{\Bbb U},

A divide-and-conquer algorithm for min-cost perfect matching in the plane

the set cover problem is to find the smallest subcollection

Decomposing Coverings and the Planar Sensor Cover Problem

C\subset S