4,313 research outputs found
Facility Location in Evolving Metrics
Understanding the dynamics of evolving social or infrastructure networks is a
challenge in applied areas such as epidemiology, viral marketing, or urban
planning. During the past decade, data has been collected on such networks but
has yet to be fully analyzed. We propose to use information on the dynamics of
the data to find stable partitions of the network into groups. For that
purpose, we introduce a time-dependent, dynamic version of the facility
location problem, that includes a switching cost when a client's assignment
changes from one facility to another. This might provide a better
representation of an evolving network, emphasizing the abrupt change of
relationships between subjects rather than the continuous evolution of the
underlying network. We show that in realistic examples this model yields indeed
better fitting solutions than optimizing every snapshot independently. We
present an -approximation algorithm and a matching hardness result,
where is the number of clients and the number of time steps. We also
give an other algorithms with approximation ratio for the variant
where one pays at each time step (leasing) for each open facility
Approximate Clustering via Metric Partitioning
In this paper we consider two metric covering/clustering problems -
\textit{Minimum Cost Covering Problem} (MCC) and -clustering. In the MCC
problem, we are given two point sets (clients) and (servers), and a
metric on . We would like to cover the clients by balls centered at
the servers. The objective function to minimize is the sum of the -th
power of the radii of the balls. Here is a parameter of the
problem (but not of a problem instance). MCC is closely related to the
-clustering problem. The main difference between -clustering and MCC is
that in -clustering one needs to select balls to cover the clients.
For any \eps > 0, we describe quasi-polynomial time (1 + \eps)
approximation algorithms for both of the problems. However, in case of
-clustering the algorithm uses (1 + \eps)k balls. Prior to our work, a
and a approximation were achieved by
polynomial-time algorithms for MCC and -clustering, respectively, where is an absolute constant. These two problems are thus interesting examples of
metric covering/clustering problems that admit (1 + \eps)-approximation
(using (1+\eps)k balls in case of -clustering), if one is willing to
settle for quasi-polynomial time. In contrast, for the variant of MCC where
is part of the input, we show under standard assumptions that no
polynomial time algorithm can achieve an approximation factor better than
for .Comment: 19 page
Minimum-Cost Coverage of Point Sets by Disks
We consider a class of geometric facility location problems in which the goal
is to determine a set X of disks given by their centers (t_j) and radii (r_j)
that cover a given set of demand points Y in the plane at the smallest possible
cost. We consider cost functions of the form sum_j f(r_j), where f(r)=r^alpha
is the cost of transmission to radius r. Special cases arise for alpha=1 (sum
of radii) and alpha=2 (total area); power consumption models in wireless
network design often use an exponent alpha>2. Different scenarios arise
according to possible restrictions on the transmission centers t_j, which may
be constrained to belong to a given discrete set or to lie on a line, etc. We
obtain several new results, including (a) exact and approximation algorithms
for selecting transmission points t_j on a given line in order to cover demand
points Y in the plane; (b) approximation algorithms (and an algebraic
intractability result) for selecting an optimal line on which to place
transmission points to cover Y; (c) a proof of NP-hardness for a discrete set
of transmission points in the plane and any fixed alpha>1; and (d) a
polynomial-time approximation scheme for the problem of computing a minimum
cost covering tour (MCCT), in which the total cost is a linear combination of
the transmission cost for the set of disks and the length of a tour/path that
connects the centers of the disks.Comment: 10 pages, 4 figures, Latex, to appear in ACM Symposium on
Computational Geometry 200
Dynamic Clustering to Minimize the Sum of Radii
In this paper, we study the problem of opening centers to cluster a set of clients in a metric space so as to minimize the sum of the costs of the centers and of the cluster radii, in a dynamic environment where clients arrive and depart, and the solution must be updated efficiently while remaining competitive with respect to the current optimal solution. We call this dynamic sum-of-radii clustering problem.
We present a data structure that maintains a solution whose cost is within a constant factor of the cost of an optimal solution in metric spaces with bounded doubling dimension and whose worst-case update time is logarithmic in the parameters of the problem
A Constant-Factor Approximation for Multi-Covering with Disks
We consider variants of the following multi-covering problem with disks. We
are given two point sets (servers) and (clients) in the plane, a
coverage function , and a constant . Centered at each server is a single disk whose radius we are free to
set. The requirement is that each client be covered by at least
of the server disks. The objective function we wish to minimize is
the sum of the -th powers of the disk radii. We present a polynomial
time algorithm for this problem achieving an approximation
Maximum gradient embeddings and monotone clustering
Let (X,d_X) be an n-point metric space. We show that there exists a
distribution D over non-contractive embeddings into trees f:X-->T such that for
every x in X, the expectation with respect to D of the maximum over y in X of
the ratio d_T(f(x),f(y)) / d_X(x,y) is at most C (log n)^2, where C is a
universal constant. Conversely we show that the above quadratic dependence on
log n cannot be improved in general. Such embeddings, which we call maximum
gradient embeddings, yield a framework for the design of approximation
algorithms for a wide range of clustering problems with monotone costs,
including fault-tolerant versions of k-median and facility location.Comment: 25 pages, 2 figures. Final version, minor revision of the previous
one. To appear in "Combinatorica
Unifying Sparsest Cut, Cluster Deletion, and Modularity Clustering Objectives with Correlation Clustering
Graph clustering, or community detection, is the task of identifying groups
of closely related objects in a large network. In this paper we introduce a new
community-detection framework called LambdaCC that is based on a specially
weighted version of correlation clustering. A key component in our methodology
is a clustering resolution parameter, , which implicitly controls the
size and structure of clusters formed by our framework. We show that, by
increasing this parameter, our objective effectively interpolates between two
different strategies in graph clustering: finding a sparse cut and forming
dense subgraphs. Our methodology unifies and generalizes a number of other
important clustering quality functions including modularity, sparsest cut, and
cluster deletion, and places them all within the context of an optimization
problem that has been well studied from the perspective of approximation
algorithms. Our approach is particularly relevant in the regime of finding
dense clusters, as it leads to a 2-approximation for the cluster deletion
problem. We use our approach to cluster several graphs, including large
collaboration networks and social networks
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