73 research outputs found
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
On Coloring Resilient Graphs
We introduce a new notion of resilience for constraint satisfaction problems,
with the goal of more precisely determining the boundary between NP-hardness
and the existence of efficient algorithms for resilient instances. In
particular, we study -resiliently -colorable graphs, which are those
-colorable graphs that remain -colorable even after the addition of any
new edges. We prove lower bounds on the NP-hardness of coloring resiliently
colorable graphs, and provide an algorithm that colors sufficiently resilient
graphs. We also analyze the corresponding notion of resilience for -SAT.
This notion of resilience suggests an array of open questions for graph
coloring and other combinatorial problems.Comment: Appearing in MFCS 201
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
Clustering is difficult only when it does not matter
Numerous papers ask how difficult it is to cluster data. We suggest that the
more relevant and interesting question is how difficult it is to cluster data
sets {\em that can be clustered well}. More generally, despite the ubiquity and
the great importance of clustering, we still do not have a satisfactory
mathematical theory of clustering. In order to properly understand clustering,
it is clearly necessary to develop a solid theoretical basis for the area. For
example, from the perspective of computational complexity theory the clustering
problem seems very hard. Numerous papers introduce various criteria and
numerical measures to quantify the quality of a given clustering. The resulting
conclusions are pessimistic, since it is computationally difficult to find an
optimal clustering of a given data set, if we go by any of these popular
criteria. In contrast, the practitioners' perspective is much more optimistic.
Our explanation for this disparity of opinions is that complexity theory
concentrates on the worst case, whereas in reality we only care for data sets
that can be clustered well.
We introduce a theoretical framework of clustering in metric spaces that
revolves around a notion of "good clustering". We show that if a good
clustering exists, then in many cases it can be efficiently found. Our
conclusion is that contrary to popular belief, clustering should not be
considered a hard task
On the practically interesting instances of MAXCUT
The complexity of a computational problem is traditionally quantified based
on the hardness of its worst case. This approach has many advantages and has
led to a deep and beautiful theory. However, from the practical perspective,
this leaves much to be desired. In application areas, practically interesting
instances very often occupy just a tiny part of an algorithm's space of
instances, and the vast majority of instances are simply irrelevant. Addressing
these issues is a major challenge for theoretical computer science which may
make theory more relevant to the practice of computer science.
Following Bilu and Linial, we apply this perspective to MAXCUT, viewed as a
clustering problem. Using a variety of techniques, we investigate practically
interesting instances of this problem. Specifically, we show how to solve in
polynomial time distinguished, metric, expanding and dense instances of MAXCUT
under mild stability assumptions. In particular, -stability
(which is optimal) suffices for metric and dense MAXCUT. We also show how to
solve in polynomial time -stable instances of MAXCUT,
substantially improving the best previously known result
- …