10,428 research outputs found
Approximation Schemes for Min-Sum k-Clustering
We consider the Min-Sum k-Clustering (k-MSC) problem. Given a set of points in a metric which is represented by an edge-weighted graph G = (V, E) and a parameter k, the goal is to partition the points V into k clusters such that the sum of distances between all pairs of the points within the same cluster is minimized.
The k-MSC problem is known to be APX-hard on general metrics. The best known approximation algorithms for the problem obtained by Behsaz, Friggstad, Salavatipour and Sivakumar [Algorithmica 2019] achieve an approximation ratio of O(log |V|) in polynomial time for general metrics and an approximation ratio 2+? in quasi-polynomial time for metrics with bounded doubling dimension. No approximation schemes for k-MSC (when k is part of the input) is known for any non-trivial metrics prior to our work. In fact, most of the previous works rely on the simple fact that there is a 2-approximate reduction from k-MSC to the balanced k-median problem and design approximation algorithms for the latter to obtain an approximation for k-MSC.
In this paper, we obtain the first Quasi-Polynomial Time Approximation Schemes (QPTAS) for the problem on metrics induced by graphs of bounded treewidth, graphs of bounded highway dimension, graphs of bounded doubling dimensions (including fixed dimensional Euclidean metrics), and planar and minor-free graphs. We bypass the barrier of 2 for k-MSC by introducing a new clustering problem, which we call min-hub clustering, which is a generalization of balanced k-median and is a trade off between center-based clustering problems (such as balanced k-median) and pair-wise clustering (such as Min-Sum k-clustering). We then show how one can find approximation schemes for Min-hub clustering on certain classes of metrics
Approximation Algorithms for Min-Sum k-Clustering and Balanced k-Median
We consider two closely related fundamental clustering problems in this paper. In the min-sum k-clustering one is given a metric space and has to partition the points into k clusters while minimizing the sum of pairwise distances between the points within the clusters. In the Balanced k-Median problem the instance is the same and one has to obtain a clustering into k cluster C1,..., Ck, where each cluster Ci has a center ci, while minimizing the total assignment costs for the points in the metric; here the cost of assigning a point j to a cluster Ci is equal to |Ci | times the j, cj distance in the metric. In this paper, we present an O(log n)-approximation for both these problems where n is the number of points in the metric that are to be served. This is an improvement over the O(−1 log1+ n)-approximation (for any constant > 0) obtained by Bartal, Charikar, and Raz [STOC ’01]. We also obtain a quasi-PTAS for Balanced k-Median in metrics with constant doubling dimension. As in the work of Bartal et al., our approximation for general metrics uses embeddings into tree metrics. The main technical contribution in this paper is an O(1)-approximation for Balanced k-Median in hierarchically separated trees (HSTs). Our improvement comes from a more direct dynamic programming approach that heavily exploits properties of standard HSTs. In this way, we avoid the reduction to special types of HSTs that were considered by Bartal et al., thereby avoiding an additional O(−1 log n) loss
The Bane of Low-Dimensionality Clustering
In this paper, we give a conditional lower bound of on
running time for the classic k-median and k-means clustering objectives (where
n is the size of the input), even in low-dimensional Euclidean space of
dimension four, assuming the Exponential Time Hypothesis (ETH). We also
consider k-median (and k-means) with penalties where each point need not be
assigned to a center, in which case it must pay a penalty, and extend our lower
bound to at least three-dimensional Euclidean space.
This stands in stark contrast to many other geometric problems such as the
traveling salesman problem, or computing an independent set of unit spheres.
While these problems benefit from the so-called (limited) blessing of
dimensionality, as they can be solved in time or
in d dimensions, our work shows that widely-used clustering
objectives have a lower bound of , even in dimension four.
We complete the picture by considering the two-dimensional case: we show that
there is no algorithm that solves the penalized version in time less than
, and provide a matching upper bound of .
The main tool we use to establish these lower bounds is the placement of
points on the moment curve, which takes its inspiration from constructions of
point sets yielding Delaunay complexes of high complexity
The Unreasonable Success of Local Search: Geometric Optimization
What is the effectiveness of local search algorithms for geometric problems
in the plane? We prove that local search with neighborhoods of magnitude
is an approximation scheme for the following problems in the
Euclidian plane: TSP with random inputs, Steiner tree with random inputs,
facility location (with worst case inputs), and bicriteria -median (also
with worst case inputs). The randomness assumption is necessary for TSP
On Variants of k-means Clustering
\textit{Clustering problems} often arise in the fields like data mining,
machine learning etc. to group a collection of objects into similar groups with
respect to a similarity (or dissimilarity) measure. Among the clustering
problems, specifically \textit{-means} clustering has got much attention
from the researchers. Despite the fact that -means is a very well studied
problem its status in the plane is still an open problem. In particular, it is
unknown whether it admits a PTAS in the plane. The best known approximation
bound in polynomial time is 9+\eps.
In this paper, we consider the following variant of -means. Given a set
of points in and a real , find a finite set of
points in that minimizes the quantity . For any fixed dimension , we design a local
search PTAS for this problem. We also give a "bi-criterion" local search
algorithm for -means which uses (1+\eps)k centers and yields a solution
whose cost is at most (1+\eps) times the cost of an optimal -means
solution. The algorithm runs in polynomial time for any fixed dimension.
The contribution of this paper is two fold. On the one hand, we are being
able to handle the square of distances in an elegant manner, which yields near
optimal approximation bound. This leads us towards a better understanding of
the -means problem. On the other hand, our analysis of local search might
also be useful for other geometric problems. This is important considering that
very little is known about the local search method for geometric approximation.Comment: 15 page
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