191 research outputs found
Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs
We study the family of intersection graphs of low density objects in low
dimensional Euclidean space. This family is quite general, and includes planar
graphs. We prove that such graphs have small separators. Next, we present
efficient -approximation algorithms for these graphs, for
Independent Set, Set Cover, and Dominating Set problems, among others. We also
prove corresponding hardness of approximation for some of these optimization
problems, providing a characterization of their intractability in terms of
density
Constant-Factor Approximation for TSP with Disks
We revisit the traveling salesman problem with neighborhoods (TSPN) and
present the first constant-ratio approximation for disks in the plane: Given a
set of disks in the plane, a TSP tour whose length is at most times
the optimal can be computed in time that is polynomial in . Our result is
the first constant-ratio approximation for a class of planar convex bodies of
arbitrary size and arbitrary intersections. In order to achieve a
-approximation, we reduce the traveling salesman problem with disks, up
to constant factors, to a minimum weight hitting set problem in a geometric
hypergraph. The connection between TSPN and hitting sets in geometric
hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure
On Variants of k-means Clustering
\textit{Clustering problems} often arise in the fields like data mining,
machine learning etc. to group a collection of objects into similar groups with
respect to a similarity (or dissimilarity) measure. Among the clustering
problems, specifically \textit{-means} clustering has got much attention
from the researchers. Despite the fact that -means is a very well studied
problem its status in the plane is still an open problem. In particular, it is
unknown whether it admits a PTAS in the plane. The best known approximation
bound in polynomial time is 9+\eps.
In this paper, we consider the following variant of -means. Given a set
of points in and a real , find a finite set of
points in that minimizes the quantity . For any fixed dimension , we design a local
search PTAS for this problem. We also give a "bi-criterion" local search
algorithm for -means which uses (1+\eps)k centers and yields a solution
whose cost is at most (1+\eps) times the cost of an optimal -means
solution. The algorithm runs in polynomial time for any fixed dimension.
The contribution of this paper is two fold. On the one hand, we are being
able to handle the square of distances in an elegant manner, which yields near
optimal approximation bound. This leads us towards a better understanding of
the -means problem. On the other hand, our analysis of local search might
also be useful for other geometric problems. This is important considering that
very little is known about the local search method for geometric approximation.Comment: 15 page
Simple PTAS's for families of graphs excluding a minor
We show that very simple algorithms based on local search are polynomial-time
approximation schemes for Maximum Independent Set, Minimum Vertex Cover and
Minimum Dominating Set, when the input graphs have a fixed forbidden minor.Comment: To appear in Discrete Applied Mathematic
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