6 research outputs found
Super-Fast Distributed Algorithms for Metric Facility Location
This paper presents a distributed O(1)-approximation algorithm, with
expected- running time, in the model for
the metric facility location problem on a size- clique network. Though
metric facility location has been considered by a number of researchers in
low-diameter settings, this is the first sub-logarithmic-round algorithm for
the problem that yields an O(1)-approximation in the setting of non-uniform
facility opening costs. In order to obtain this result, our paper makes three
main technical contributions. First, we show a new lower bound for metric
facility location, extending the lower bound of B\u{a}doiu et al. (ICALP 2005)
that applies only to the special case of uniform facility opening costs. Next,
we demonstrate a reduction of the distributed metric facility location problem
to the problem of computing an O(1)-ruling set of an appropriate spanning
subgraph. Finally, we present a sub-logarithmic-round (in expectation)
algorithm for computing a 2-ruling set in a spanning subgraph of a clique. Our
algorithm accomplishes this by using a combination of randomized and
deterministic sparsification.Comment: 15 pages, 2 figures. This is the full version of a paper that
appeared in ICALP 201
A Super-Fast Distributed Algorithm for Bipartite Metric Facility Location
The \textit{facility location} problem consists of a set of
\textit{facilities} , a set of \textit{clients} , an
\textit{opening cost} associated with each facility , and a
\textit{connection cost} between each facility and client
. The goal is to find a subset of facilities to \textit{open}, and to
connect each client to an open facility, so as to minimize the total facility
opening costs plus connection costs. This paper presents the first
expected-sub-logarithmic-round distributed O(1)-approximation algorithm in the
model for the \textit{metric} facility location problem on
the complete bipartite network with parts and . Our
algorithm has an expected running time of rounds, where . This result can be viewed as a continuation
of our recent work (ICALP 2012) in which we presented the first
sub-logarithmic-round distributed O(1)-approximation algorithm for metric
facility location on a \textit{clique} network. The bipartite setting presents
several new challenges not present in the problem on a clique network. We
present two new techniques to overcome these challenges. (i) In order to deal
with the problem of not being able to choose appropriate probabilities (due to
lack of adequate knowledge), we design an algorithm that performs a random walk
over a probability space and analyze the progress our algorithm makes as the
random walk proceeds. (ii) In order to deal with a problem of quickly
disseminating a collection of messages, possibly containing many duplicates,
over the bipartite network, we design a probabilistic hashing scheme that
delivers all of the messages in expected- rounds.Comment: 22 pages. This is the full version of a paper that appeared in DISC
201
An optimal maximal independent setalgorithm for bounded-independence graphs
We present a novel distributed algorithm for the maximal independent set problem (This is an extended journal version of Schneider and Wattenhofer in Twenty-seventh annual ACM SIGACT-SIGOPS symposium on principles of distributed computing, 2008). On bounded-independence graphs our deterministic algorithm finishes in O(log* n) time, n being the number of nodes. In light of Linial's Ω(log* n) lower bound our algorithm is asymptotically optimal. Furthermore, it solves the connected dominating set problem for unit disk graphs in O(log* n) time, exponentially faster than the state-of-the-art algorithm. With a new extension our algorithm also computes a δ+1 coloring and a maximal matching in O(log* n) time, where δ is the maximum degree of the grap
A Weakly-Robust PTAS for Minimum Clique Partition in Unit Disk Graphs
We consider the problem of partitioning the set of vertices of a given unit
disk graph (UDG) into a minimum number of cliques. The problem is NP-hard and
various constant factor approximations are known, with the current best ratio
of 3. Our main result is a {\em weakly robust} polynomial time approximation
scheme (PTAS) for UDGs expressed with edge-lengths, it either (i) computes a
clique partition or (ii) gives a certificate that the graph is not a UDG; for
the case (i) that it computes a clique partition, we show that it is guaranteed
to be within (1+\eps) ratio of the optimum if the input is UDG; however if
the input is not a UDG it either computes a clique partition as in case (i)
with no guarantee on the quality of the clique partition or detects that it is
not a UDG. Noting that recognition of UDG's is NP-hard even if we are given
edge lengths, our PTAS is a weakly-robust algorithm. Our algorithm can be
transformed into an O(\frac{\log^* n}{\eps^{O(1)}}) time distributed PTAS.
We consider a weighted version of the clique partition problem on vertex
weighted UDGs that generalizes the problem. We note some key distinctions with
the unweighted version, where ideas useful in obtaining a PTAS breakdown. Yet,
surprisingly, it admits a (2+\eps)-approximation algorithm for the weighted
case where the graph is expressed, say, as an adjacency matrix. This improves
on the best known 8-approximation for the {\em unweighted} case for UDGs
expressed in standard form.Comment: 21 pages, 9 figure
Local and online algorithms for facility location
Diese Arbeit beschäftigt sich mit dem Facility Location Problem. Dies ist ein Optimierungsproblem, bei dem festgelegt werden muss an welchen Positionen Ressourcen zur Verfügung gestellt werden, so dass diese von Nutzern gut erreicht werden können. Es sollen dabei Kosten minimiert werden, die zum einen durch Bereitstellung von Ressourcen und zum anderen durch Verbindungskosten zwischen Nutzern und Ressourcen entstehen. In dieser Arbeit werden drei Varianten des Problems modelliert und neue Algorithmen für sie entwickelt und bezüglich ihres Approximationsfaktors und ihrer Laufzeit analysiert. Jede dieser drei untersuchten Varianten hat einen besonderen Schwerpunkt. Bei der ersten Varianten handelt es sich um ein Online Problem, da hier die Eingabe nicht von Anfang an bekannt ist, sondern Schritt für Schritt enthüllt wird. Die Schwierigkeit hierbei besteht darin unwiderrufliche Entscheidungen treffen zu müssen ohne dabei die Zukunft zu kennen und trotzdem eine zu jeder Zeit gute Lösung angeben zu können. Der Schwerpunkt der zweiten Variante liegt auf Lokalität. Hier soll eine Lösung verteilt und nur mit Hilfe von lokalen Information berechnet werden. Schließlich beschäftigt sich die dritte Variante mit einer verteilten Berechnung, bei welcher nur eine stark beschränkte Datenmenge verschickt werden darf und dabei trotzdem ein sehr guter Approximationsfaktor erreicht werden muss. Die bei der Analyse der Approximationsfaktoren bzw. der Kompetitivität verwendeten TecTechniken basieren zum großen Teil auf Abschätzung der primalen Lösung mit Hilfe einer Lösung des zugehörigen dualen Problems. Für die Modellierung von Lokalität wird das weitverbreitete LOCAL Modell verwendet. In diesem Modell werden für die Algorithmen subpolynomielle obere Laufzeitschranken gezeigt.The topic of this thesis is approximation and online algorithms for an optimization problem known as Facility Location. This problem, or one of its many variants, arises as a sub problem in many practical applications, and is thus of significant importance in the field of Operations Research. Furthermore, it is also one of the most studied optimization problems in theoretical computer science with hundreds of research papers published during the last decades. In this thesis, we focus on the theoretical aspects of Facility Location by designing and analyzing approximation and online algorithms. Our algorithms deal with three distinct scenarios in which Facility Location occurs: (i) networks that are exposed to perpetual changes, (ii) wireless sensor networks with strong locality constraints, and (iii) distributed settings where the focus lies, first and foremost, on the quality of the computed approximation. Chapter 2 covers Scenario (i). It presents an online algorithm designed for a highly dynamic network where additional nodes are perpetually added. The difficulty here is that these new nodes' requests have to be handled efficiently without any knowledge of the network's future development. Scenario (ii) is considered in Chapter 3. Two distributed algorithms for wireless sensor networks are presented here. Due to the nodes' limited communication range, locality is of high importance in this scenario. Additional aspects like inaccurate measurement data, power consumption, and dynamics are also taken into account. Finally, Scenario (iii) is considered in Chapter 4. Our objective here is to distributedly compute a solution with an approximation ratio that is as close as possible to the best achievable ratio. In order to accomplish this, we allow, compared to Scenario (ii), a higher running time, but still require that the algorithm terminates in sub-linear time.Tag der Verteidigung: 28.10.2013Paderborn, Univ., Diss., 201
Finding Facilities Fast
Clustering can play a critical role in increasing the performance and lifetime of wireless networks. The facility location problem is a general abstraction of the clustering problem and this paper presents the first constant-factor approximation algorithm for the facility location problem on unit disk graphs (UDGs), a commonly used model for wireless networks. In this version of the problem, connection costs are not metric, i.e., they do not satisfy the triangle inequality, because connecting to a non-neighbor costs ∞. In non-metric settings the best approximation algorithms guarantee an O(log n)-factor approximation, but we are able to use structural properties of UDGs to obtain a constant-factor approximation. Our approach combines ideas from the primal-dual algorithm for facility location due to Jain and Vazirani (JACM, 2001) with recent results on the weighted minimum dominating set problem for UDGs (Huang et al., J. Comb. Opt., 2008). We then show that the facility location problem on UDGs is inherently local and one can solve local subproblems independently and combine the solutions in a simple way to obtain a good solution to the overall problem. This leads to a distributed version of our algorithm in the LOCAL model that runs in constant rounds and still yields a constant-factor approximation. Even if the UDG is specified without geometry, we are able to combine recent results on maximal independent sets and clique partitioning of UDGs, to obtain an O(log n)-approximation that runs in O(log ∗ n) rounds