Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: FPT Approximation for Fair Minimum-Load Clustering   In this paper, we consider the Minimum-Load $k$-Clustering/Facility Location
(MLkC) problem where we are given a set $P$ of $n$ points in a metric space
that we have to cluster and an integer $k$ that denotes the number of clusters.
Additionally, we are given a set $F$ of cluster centers in the same metric
space. The goal is to select a set $C\subseteq F$ of $k$ centers and assign
each point in $P$ to a center in $C$, such that the maximum load over all
centers is minimized. Here the load of a center is the sum of the distances
between it and the points assigned to it.
  Although clustering/facility location problems have a rich literature, the
minimum-load objective is not studied substantially, and hence MLkC has
remained a poorly understood problem. More interestingly, the problem is
notoriously hard even in some special cases including the one in line metrics
as shown by Ahmadian et al. [ACM Trans. Algo. 2018]. They also show
APX-hardness of the problem in the plane. On the other hand, the best-known
approximation factor for MLkC is $O(k)$, even in the plane.
  In this work, we study a fair version of MLkC inspired by the work of
Chierichetti et al. [NeurIPS, 2017], which generalizes MLkC. Here the input
points are colored by one of the $\ell$ colors denoting the group they belong
to. MLkC is the special case with $\ell=1$. Considering this problem, we are
able to obtain a $3$-approximation in $f(k,\ell)\cdot n^{O(1)}$ time. Also, our
scheme leads to an improved $(1 + \epsilon)$-approximation in case of Euclidean
norm, and in this case, the running time depends only polynomially on the
dimension $d$. Our results imply the same approximations for MLkC with running
time $f(k)\cdot n^{O(1)}$, achieving the first constant approximations for this
problem in general and Euclidean metric spaces.

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The categories of this paper are: dg-ga
21
The categories of this paper are: adap-org
7
The categories of this paper are: hep-th
15
The categories of this paper are: nucl-ex
4
The categories of this paper are: cond-mat
9
The categories of this paper are: nlin
3
The categories of this paper are: physics
6
The categories of this paper are: astro-ph
10
The categories of this paper are: q-bio
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The categories of this paper are: bayes-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: alg-geom
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The categories of this paper are: comp-gas
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The categories of this paper are: stat
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The categories of this paper are: solv-int
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The categories of this paper are: chao-dyn