Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Improved Approximation Algorithms for Geometric Set Cover   Given a collection S of subsets of some set U, and M a subset of U, the set
cover problem is to find the smallest subcollection C of S such that M is a
subset of the union of the sets in C. While the general problem is NP-hard to
solve, even approximately, here we consider some geometric special cases, where
usually U = R^d. Extending prior results, we show that approximation algorithms
with provable performance exist, under a certain general condition: that for a
random subset R of S and function f(), there is a decomposition of the portion
of U not covered by R into an expected f(|R|) regions, each region of a
particular simple form. We show that under this condition, a cover of size
O(f(|C|)) can be found. Our proof involves the generalization of shallow
cuttings to more general geometric situations. We obtain constant-factor
approximation algorithms for covering by unit cubes in R^3, for guarding a
one-dimensional terrain, and for covering by similar-sized fat triangles in
R^2. We also obtain improved approximation guarantees for fat triangles, of
arbitrary size, and for a class of fat objects.

18
The categories of this paper are: econ
13
The categories of this paper are: quant-ph
21
The categories of this paper are: adap-org
33
The categories of this paper are: plasm-ph
26
The categories of this paper are: bayes-an
4
The categories of this paper are: cond-mat
2
The categories of this paper are: math
22
The categories of this paper are: chao-dyn
23
The categories of this paper are: solv-int
25
The categories of this paper are: patt-sol
31
The categories of this paper are: ao-sci
35
The categories of this paper are: mtrl-th
6
The categories of this paper are: astro-ph
30
The categories of this paper are: q-alg
37
The categories of this paper are: supr-con
7
The categories of this paper are: hep-th