584 research outputs found
Improved Bounds for Online Preemptive Matching
When designing a preemptive online algorithm for the maximum matching
problem, we wish to maintain a valid matching M while edges of the underlying
graph are presented one after the other. When presented with an edge e, the
algorithm should decide whether to augment the matching M by adding e (in which
case e may be removed later on) or to keep M in its current form without adding
e (in which case e is lost for good). The objective is to eventually hold a
matching M with maximum weight.
The main contribution of this paper is to establish new lower and upper
bounds on the competitive ratio achievable by preemptive online algorithms:
1. We provide a lower bound of 1+ln 2~1.693 on the competitive ratio of any
randomized algorithm for the maximum cardinality matching problem, thus
improving on the currently best known bound of e/(e-1)~1.581 due to Karp,
Vazirani, and Vazirani [STOC'90].
2. We devise a randomized algorithm that achieves an expected competitive
ratio of 5.356 for maximum weight matching. This finding demonstrates the power
of randomization in this context, showing how to beat the tight bound of 3
+2\sqrt{2}~5.828 for deterministic algorithms, obtained by combining the 5.828
upper bound of McGregor [APPROX'05] and the recent 5.828 lower bound of
Varadaraja [ICALP'11]
On Randomized Algorithms for Matching in the Online Preemptive Model
We investigate the power of randomized algorithms for the maximum cardinality
matching (MCM) and the maximum weight matching (MWM) problems in the online
preemptive model. In this model, the edges of a graph are revealed one by one
and the algorithm is required to always maintain a valid matching. On seeing an
edge, the algorithm has to either accept or reject the edge. If accepted, then
the adjacent edges are discarded. The complexity of the problem is settled for
deterministic algorithms.
Almost nothing is known for randomized algorithms. A lower bound of
is known for MCM with a trivial upper bound of . An upper bound of
is known for MWM. We initiate a systematic study of the same in this paper with
an aim to isolate and understand the difficulty. We begin with a primal-dual
analysis of the deterministic algorithm due to McGregor. All deterministic
lower bounds are on instances which are trees at every step. For this class of
(unweighted) graphs we present a randomized algorithm which is
-competitive. The analysis is a considerable extension of the
(simple) primal-dual analysis for the deterministic case. The key new technique
is that the distribution of primal charge to dual variables depends on the
"neighborhood" and needs to be done after having seen the entire input. The
assignment is asymmetric: in that edges may assign different charges to the two
end-points. Also the proof depends on a non-trivial structural statement on the
performance of the algorithm on the input tree.
The other main result of this paper is an extension of the deterministic
lower bound of Varadaraja to a natural class of randomized algorithms which
decide whether to accept a new edge or not using independent random choices
Online Service with Delay
In this paper, we introduce the online service with delay problem. In this
problem, there are points in a metric space that issue service requests
over time, and a server that serves these requests. The goal is to minimize the
sum of distance traveled by the server and the total delay in serving the
requests. This problem models the fundamental tradeoff between batching
requests to improve locality and reducing delay to improve response time, that
has many applications in operations management, operating systems, logistics,
supply chain management, and scheduling.
Our main result is to show a poly-logarithmic competitive ratio for the
online service with delay problem. This result is obtained by an algorithm that
we call the preemptive service algorithm. The salient feature of this algorithm
is a process called preemptive service, which uses a novel combination of
(recursive) time forwarding and spatial exploration on a metric space. We hope
this technique will be useful for related problems such as reordering buffer
management, online TSP, vehicle routing, etc. We also generalize our results to
servers.Comment: 30 pages, 11 figures, Appeared in 49th ACM Symposium on Theory of
Computing (STOC), 201
Improved approximation guarantees for weighted matching in the semi-streaming model
We study the maximum weight matching problem in the semi-streaming model, and
improve on the currently best one-pass algorithm due to Zelke (Proc. of
STACS2008, pages 669-680) by devising a deterministic approach whose
performance guarantee is 4.91+epsilon. In addition, we study preemptive online
algorithms, a sub-class of one-pass algorithms where we are only allowed to
maintain a feasible matching in memory at any point in time. All known results
prior to Zelke's belong to this sub-class. We provide a lower bound of 4.967 on
the competitive ratio of any such deterministic algorithm, and hence show that
future improvements will have to store in memory a set of edges which is not
necessarily a feasible matching
Streaming Algorithms for Submodular Function Maximization
We consider the problem of maximizing a nonnegative submodular set function
subject to a -matchoid
constraint in the single-pass streaming setting. Previous work in this context
has considered streaming algorithms for modular functions and monotone
submodular functions. The main result is for submodular functions that are {\em
non-monotone}. We describe deterministic and randomized algorithms that obtain
a -approximation using -space, where is
an upper bound on the cardinality of the desired set. The model assumes value
oracle access to and membership oracles for the matroids defining the
-matchoid constraint.Comment: 29 pages, 7 figures, extended abstract to appear in ICALP 201
Probabilistic alternatives for competitive analysis
In the last 20 years competitive analysis has become the main tool for analyzing the quality of online algorithms. Despite of this, competitive analysis has also been criticized: it sometimes cannot discriminate between algorithms that exhibit significantly different empirical behavior or it even favors an algorithm that is worse from an empirical point of view. Therefore, there have been several approaches to circumvent these drawbacks. In this survey, we discuss probabilistic alternatives for competitive analysis.operations research and management science;
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