6,645 research outputs found
k-Center Clustering Under Perturbation Resilience
The k-center problem is a canonical and long-studied facility location and clustering problem with many applications in both its symmetric and asymmetric forms. Both versions of the problem have tight approximation factors on worst case instances: a 2-approximation for symmetric kcenter and an O(log*(k))-approximation for the asymmetric version. Therefore to improve on these ratios, one must go beyond the worst case.
In this work, we take this approach and provide strong positive results both for the asymmetric and symmetric k-center problems under a very natural input stability (promise) condition called alpha-perturbation resilience [Bilu Linial, 2012], which states that the optimal solution does not change under any alpha-factor perturbation to the input distances. We show that by assuming 2-perturbation resilience, the exact solution for the asymmetric k-center problem can be found in polynomial time. To our knowledge, this is the first problem that is hard to approximate to any constant factor in the worst case, yet can be optimally solved in polynomial time under perturbation resilience for a constant value of alpha. Furthermore, we prove our result is tight by showing symmetric k-center under (2-epsilon)-perturbation resilience is hard unless NP=RP.
This is the first tight result for any problem under perturbation resilience, i.e., this is the first time the exact value of alpha for which the problem switches from being NP-hard to efficiently computable has been found.
Our results illustrate a surprising relationship between symmetric and asymmetric k-center instances under perturbation resilience. Unlike approximation ratio, for which symmetric k-center is easily solved to a factor of 2 but asymmetric k-center cannot be approximated to any constant factor, both symmetric and asymmetric k-center can be solved optimally under resilience
to 2-perturbations
On Perturbation Resilience of Non-Uniform k-Center
The Non-Uniform -center (NUkC) problem has recently been formulated by
Chakrabarty, Goyal and Krishnaswamy [ICALP, 2016] as a generalization of the
classical -center clustering problem. In NUkC, given a set of points
in a metric space and non-negative numbers , the goal
is to find the minimum dilation and to choose balls centered at
the points of with radius for , such that
all points of are contained in the union of the chosen balls. They showed
that the problem is NP-hard to approximate within any factor even in tree
metrics. On the other hand, they designed a "bi-criteria" constant
approximation algorithm that uses a constant times balls. Surprisingly, no
true approximation is known even in the special case when the 's belong to
a fixed set of size 3. In this paper, we study the NUkC problem under
perturbation resilience, which was introduced by Bilu and Linial
[Combinatorics, Probability and Computing, 2012]. We show that the problem
under 2-perturbation resilience is polynomial time solvable when the 's
belong to a constant sized set. However, we show that perturbation resilience
does not help in the general case. In particular, our findings imply that even
with perturbation resilience one cannot hope to find any "good" approximation
for the problem.Comment: 20 pages, 5 figure
Center-based Clustering under Perturbation Stability
Clustering under most popular objective functions is NP-hard, even to
approximate well, and so unlikely to be efficiently solvable in the worst case.
Recently, Bilu and Linial \cite{Bilu09} suggested an approach aimed at
bypassing this computational barrier by using properties of instances one might
hope to hold in practice. In particular, they argue that instances in practice
should be stable to small perturbations in the metric space and give an
efficient algorithm for clustering instances of the Max-Cut problem that are
stable to perturbations of size . In addition, they conjecture that
instances stable to as little as O(1) perturbations should be solvable in
polynomial time. In this paper we prove that this conjecture is true for any
center-based clustering objective (such as -median, -means, and
-center). Specifically, we show we can efficiently find the optimal
clustering assuming only stability to factor-3 perturbations of the underlying
metric in spaces without Steiner points, and stability to factor
perturbations for general metrics. In particular, we show for such instances
that the popular Single-Linkage algorithm combined with dynamic programming
will find the optimal clustering. We also present NP-hardness results under a
weaker but related condition
Certified Algorithms: Worst-Case Analysis and Beyond
In this paper, we introduce the notion of a certified algorithm. Certified algorithms provide worst-case and beyond-worst-case performance guarantees. First, a ?-certified algorithm is also a ?-approximation algorithm - it finds a ?-approximation no matter what the input is. Second, it exactly solves ?-perturbation-resilient instances (?-perturbation-resilient instances model real-life instances). Additionally, certified algorithms have a number of other desirable properties: they solve both maximization and minimization versions of a problem (e.g. Max Cut and Min Uncut), solve weakly perturbation-resilient instances, and solve optimization problems with hard constraints.
In the paper, we define certified algorithms, describe their properties, present a framework for designing certified algorithms, provide examples of certified algorithms for Max Cut/Min Uncut, Minimum Multiway Cut, k-medians and k-means. We also present some negative results
Large-scale inference and graph theoretical analysis of gene-regulatory networks in B. stubtilis
We present the methods and results of a two-stage modeling process that
generates candidate gene-regulatory networks of the bacterium B. subtilis from
experimentally obtained, yet mathematically underdetermined microchip array
data. By employing a computational, linear correlative procedure to generate
these networks, and by analyzing the networks from a graph theoretical
perspective, we are able to verify the biological viability of our inferred
networks, and we demonstrate that our networks' graph theoretical properties
are remarkably similar to those of other biological systems. In addition, by
comparing our inferred networks to those of a previous, noisier implementation
of the linear inference process [17], we are able to identify trends in graph
theoretical behavior that occur both in our networks as well as in their
perturbed counterparts. These commonalities in behavior at multiple levels of
complexity allow us to ascertain the level of complexity to which our process
is robust to noise.Comment: 22 pages, 4 figures, accepted for publication in Physica A (2006
- …