Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Better Guarantees for k-Means and Euclidean k-Median by Primal-Dual
  Algorithms   Clustering is a classic topic in optimization with $k$-means being one of the
most fundamental such problems. In the absence of any restrictions on the
input, the best known algorithm for $k$-means with a provable guarantee is a
simple local search heuristic yielding an approximation guarantee of
$9+\epsilon$, a ratio that is known to be tight with respect to such methods.
  We overcome this barrier by presenting a new primal-dual approach that allows
us to (1) exploit the geometric structure of $k$-means and (2) to satisfy the
hard constraint that at most $k$ clusters are selected without deteriorating
the approximation guarantee. Our main result is a $6.357$-approximation
algorithm with respect to the standard LP relaxation. Our techniques are quite
general and we also show improved guarantees for the general version of
$k$-means where the underlying metric is not required to be Euclidean and for
$k$-median in Euclidean metrics.

13
The categories of this paper are: quant-ph
11
The categories of this paper are: hep-lat
15
The categories of this paper are: nucl-ex
22
The categories of this paper are: chao-dyn
2
The categories of this paper are: math
24
The categories of this paper are: dg-ga
35
The categories of this paper are: mtrl-th
3
The categories of this paper are: physics
16
The categories of this paper are: stat
20
The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
32
The categories of this paper are: atom-ph
25
The categories of this paper are: patt-sol
31
The categories of this paper are: ao-sci
36
The categories of this paper are: cmp-lg
26
The categories of this paper are: bayes-an