18,207 research outputs found
Online unit clustering in higher dimensions
We revisit the online Unit Clustering and Unit Covering problems in higher
dimensions: Given a set of points in a metric space, that arrive one by
one, Unit Clustering asks to partition the points into the minimum number of
clusters (subsets) of diameter at most one; while Unit Covering asks to cover
all points by the minimum number of balls of unit radius. In this paper, we
work in using the norm.
We show that the competitive ratio of any online algorithm (deterministic or
randomized) for Unit Clustering must depend on the dimension . We also give
a randomized online algorithm with competitive ratio for Unit
Clustering}of integer points (i.e., points in , , under norm). We show that the competitive ratio of
any deterministic online algorithm for Unit Covering is at least . This
ratio is the best possible, as it can be attained by a simple deterministic
algorithm that assigns points to a predefined set of unit cubes. We complement
these results with some additional lower bounds for related problems in higher
dimensions.Comment: 15 pages, 4 figures. A preliminary version appeared in the
Proceedings of the 15th Workshop on Approximation and Online Algorithms (WAOA
2017
Partitioning networks into cliques: a randomized heuristic approach
In the context of community detection in social networks, the term community can be grounded in the strict way that simply everybody should know each other within the community. We consider the corresponding community detection problem. We search for a partitioning of a network into the minimum number of non-overlapping cliques, such that the cliques cover all vertices. This problem is called the clique covering problem (CCP) and is one of the classical NP-hard problems. For CCP, we propose a randomized heuristic approach. To construct a high quality solution to CCP, we present an iterated greedy (IG) algorithm. IG can also be combined with a heuristic used to determine how far the algorithm is from the optimum in the worst case. Randomized local search (RLS) for maximum independent set was proposed to find such a bound. The experimental results of IG and the bounds obtained by RLS indicate that IG is a very suitable technique for solving CCP in real-world graphs. In addition, we summarize our basic rigorous results, which were developed for analysis of IG and understanding of its behavior on several relevant graph classes
Lagrangian Relaxation and Partial Cover
Lagrangian relaxation has been used extensively in the design of
approximation algorithms. This paper studies its strengths and limitations when
applied to Partial Cover.Comment: 20 pages, extended abstract appeared in STACS 200
Robust and MaxMin Optimization under Matroid and Knapsack Uncertainty Sets
Consider the following problem: given a set system (U,I) and an edge-weighted
graph G = (U, E) on the same universe U, find the set A in I such that the
Steiner tree cost with terminals A is as large as possible: "which set in I is
the most difficult to connect up?" This is an example of a max-min problem:
find the set A in I such that the value of some minimization (covering) problem
is as large as possible.
In this paper, we show that for certain covering problems which admit good
deterministic online algorithms, we can give good algorithms for max-min
optimization when the set system I is given by a p-system or q-knapsacks or
both. This result is similar to results for constrained maximization of
submodular functions. Although many natural covering problems are not even
approximately submodular, we show that one can use properties of the online
algorithm as a surrogate for submodularity.
Moreover, we give stronger connections between max-min optimization and
two-stage robust optimization, and hence give improved algorithms for robust
versions of various covering problems, for cases where the uncertainty sets are
given by p-systems and q-knapsacks.Comment: 17 pages. Preliminary version combining this paper and
http://arxiv.org/abs/0912.1045 appeared in ICALP 201
Truthful Facility Assignment with Resource Augmentation: An Exact Analysis of Serial Dictatorship
We study the truthful facility assignment problem, where a set of agents with
private most-preferred points on a metric space are assigned to facilities that
lie on the metric space, under capacity constraints on the facilities. The goal
is to produce such an assignment that minimizes the social cost, i.e., the
total distance between the most-preferred points of the agents and their
corresponding facilities in the assignment, under the constraint of
truthfulness, which ensures that agents do not misreport their most-preferred
points.
We propose a resource augmentation framework, where a truthful mechanism is
evaluated by its worst-case performance on an instance with enhanced facility
capacities against the optimal mechanism on the same instance with the original
capacities. We study a very well-known mechanism, Serial Dictatorship, and
provide an exact analysis of its performance. Although Serial Dictatorship is a
purely combinatorial mechanism, our analysis uses linear programming; a linear
program expresses its greedy nature as well as the structure of the input, and
finds the input instance that enforces the mechanism have its worst-case
performance. Bounding the objective of the linear program using duality
arguments allows us to compute tight bounds on the approximation ratio. Among
other results, we prove that Serial Dictatorship has approximation ratio
when the capacities are multiplied by any integer . Our
results suggest that even a limited augmentation of the resources can have
wondrous effects on the performance of the mechanism and in particular, the
approximation ratio goes to 1 as the augmentation factor becomes large. We
complement our results with bounds on the approximation ratio of Random Serial
Dictatorship, the randomized version of Serial Dictatorship, when there is no
resource augmentation
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