48 research outputs found
The Hardness of Approximation of Euclidean k-means
The Euclidean -means problem is a classical problem that has been
extensively studied in the theoretical computer science, machine learning and
the computational geometry communities. In this problem, we are given a set of
points in Euclidean space , and the goal is to choose centers in
so that the sum of squared distances of each point to its nearest center
is minimized. The best approximation algorithms for this problem include a
polynomial time constant factor approximation for general and a
-approximation which runs in time . At
the other extreme, the only known computational complexity result for this
problem is NP-hardness [ADHP'09]. The main difficulty in obtaining hardness
results stems from the Euclidean nature of the problem, and the fact that any
point in can be a potential center. This gap in understanding left open
the intriguing possibility that the problem might admit a PTAS for all .
In this paper we provide the first hardness of approximation for the
Euclidean -means problem. Concretely, we show that there exists a constant
such that it is NP-hard to approximate the -means objective
to within a factor of . We show this via an efficient reduction
from the vertex cover problem on triangle-free graphs: given a triangle-free
graph, the goal is to choose the fewest number of vertices which are incident
on all the edges. Additionally, we give a proof that the current best hardness
results for vertex cover can be carried over to triangle-free graphs. To show
this we transform , a known hard vertex cover instance, by taking a graph
product with a suitably chosen graph , and showing that the size of the
(normalized) maximum independent set is almost exactly preserved in the product
graph using a spectral analysis, which might be of independent interest
Tight Analysis of a Multiple-Swap Heuristic for Budgeted Red-Blue Median
Budgeted Red-Blue Median is a generalization of classic -Median in that
there are two sets of facilities, say and , that can
be used to serve clients located in some metric space. The goal is to open
facilities in and facilities in for
some given bounds and connect each client to their nearest open
facility in a way that minimizes the total connection cost.
We extend work by Hajiaghayi, Khandekar, and Kortsarz [2012] and show that a
multiple-swap local search heuristic can be used to obtain a
-approximation for Budgeted Red-Blue Median for any constant
. This is an improvement over their single swap analysis and
beats the previous best approximation guarantee of 8 by Swamy [2014].
We also present a matching lower bound showing that for every ,
there are instances of Budgeted Red-Blue Median with local optimum solutions
for the -swap heuristic whose cost is
times the optimum solution cost. Thus, our analysis is tight up to the lower
order terms. In particular, for any we show the single-swap
heuristic admits local optima whose cost can be as bad as times
the optimum solution cost
An Improved Approximation Algorithm for the Hard Uniform Capacitated k-median Problem
In the -median problem, given a set of locations, the goal is to select a
subset of at most centers so as to minimize the total cost of connecting
each location to its nearest center. We study the uniform hard capacitated
version of the -median problem, in which each selected center can only serve
a limited number of locations.
Inspired by the algorithm of Charikar, Guha, Tardos and Shmoys, we give a
-approximation algorithm for this problem with increasing the
capacities by a factor of , which improves
the previous best -approximation algorithm proposed by Byrka,
Fleszar, Rybicki and Spoerhase violating the capacities by factor
.Comment: 19 pages, 1 figur
Probabilistic Analysis of Optimization Problems on Generalized Random Shortest Path Metrics
Simple heuristics often show a remarkable performance in practice for
optimization problems. Worst-case analysis often falls short of explaining this
performance. Because of this, "beyond worst-case analysis" of algorithms has
recently gained a lot of attention, including probabilistic analysis of
algorithms.
The instances of many optimization problems are essentially a discrete metric
space. Probabilistic analysis for such metric optimization problems has
nevertheless mostly been conducted on instances drawn from Euclidean space,
which provides a structure that is usually heavily exploited in the analysis.
However, most instances from practice are not Euclidean. Little work has been
done on metric instances drawn from other, more realistic, distributions. Some
initial results have been obtained by Bringmann et al. (Algorithmica, 2013),
who have used random shortest path metrics on complete graphs to analyze
heuristics.
The goal of this paper is to generalize these findings to non-complete
graphs, especially Erd\H{o}s-R\'enyi random graphs. A random shortest path
metric is constructed by drawing independent random edge weights for each edge
in the graph and setting the distance between every pair of vertices to the
length of a shortest path between them with respect to the drawn weights. For
such instances, we prove that the greedy heuristic for the minimum distance
maximum matching problem, the nearest neighbor and insertion heuristics for the
traveling salesman problem, and a trivial heuristic for the -median problem
all achieve a constant expected approximation ratio. Additionally, we show a
polynomial upper bound for the expected number of iterations of the 2-opt
heuristic for the traveling salesman problem.Comment: An extended abstract appeared in the proceedings of WALCOM 201
An Approximation Algorithm for Multi Allocation Hub Location Problems
The multi allocation p-hub median problem (MApHM), the multi allocation
uncapacitated hub location problem (MAuHLP) and the multi allocation p-hub
location problem (MApHLP) are common hub location problems with several
practical applications. HLPs aim to construct a network for routing tasks
between different locations. Specifically, a set of hubs must be chosen and
each routing must be performed using one or two hubs as stopovers. The costs
between two hubs are discounted. The objective is to minimize the total
transportation cost in the MApHM and additionally to minimize the set-up costs
for the hubs in the MAuHLP and MApHLP. In this paper, an approximation
algorithm to solve these problems is developed, which improves the
approximation bound for MApHM to 3.451, for MAuHLP to 2.173 and for MApHLP to
4.552 when combined with the algorithm of Benedito & Pedrosa.
The proposed algorithm is capable of solving much bigger instances than any
exact algorithm in the literature. New benchmark instances have been created
and published for evaluation, such that HLP algorithms can be tested and
compared on huge instances. The proposed algorithm performs on most instances
better than the algorithm of Benedito & Pedrosa, which was the only known
approximation algorithm for these problems by now
The Non-Uniform k-Center Problem
In this paper, we introduce and study the Non-Uniform k-Center problem
(NUkC). Given a finite metric space and a collection of balls of radii
, the NUkC problem is to find a placement of their
centers on the metric space and find the minimum dilation , such that
the union of balls of radius around the th center covers
all the points in . This problem naturally arises as a min-max vehicle
routing problem with fleets of different speeds.
The NUkC problem generalizes the classic -center problem when all the
radii are the same (which can be assumed to be after scaling). It also
generalizes the -center with outliers (kCwO) problem when there are
balls of radius and balls of radius . There are -approximation
and -approximation algorithms known for these problems respectively; the
former is best possible unless P=NP and the latter remains unimproved for 15
years.
We first observe that no -approximation is to the optimal dilation is
possible unless P=NP, implying that the NUkC problem is more non-trivial than
the above two problems. Our main algorithmic result is an
-bi-criteria approximation result: we give an -approximation
to the optimal dilation, however, we may open centers of each
radii. Our techniques also allow us to prove a simple (uni-criteria), optimal
-approximation to the kCwO problem improving upon the long-standing
-factor. Our main technical contribution is a connection between the NUkC
problem and the so-called firefighter problems on trees which have been studied
recently in the TCS community.Comment: Adjusted the figur