9,323 research outputs found
Linear Programming in the Semi-streaming Model with Application to the Maximum Matching Problem
In this paper, we study linear programming based approaches to the maximum
matching problem in the semi-streaming model. The semi-streaming model has
gained attention as a model for processing massive graphs as the importance of
such graphs has increased. This is a model where edges are streamed-in in an
adversarial order and we are allowed a space proportional to the number of
vertices in a graph.
In recent years, there has been several new results in this semi-streaming
model. However broad techniques such as linear programming have not been
adapted to this model. We present several techniques to adapt and optimize
linear programming based approaches in the semi-streaming model with an
application to the maximum matching problem. As a consequence, we improve
(almost) all previous results on this problem, and also prove new results on
interesting variants
Submodular Maximization Meets Streaming: Matchings, Matroids, and More
We study the problem of finding a maximum matching in a graph given by an
input stream listing its edges in some arbitrary order, where the quantity to
be maximized is given by a monotone submodular function on subsets of edges.
This problem, which we call maximum submodular-function matching (MSM), is a
natural generalization of maximum weight matching (MWM), which is in turn a
generalization of maximum cardinality matching (MCM). We give two incomparable
algorithms for this problem with space usage falling in the semi-streaming
range---they store only edges, using working memory---that
achieve approximation ratios of in a single pass and in
passes respectively. The operations of these algorithms
mimic those of Zelke's and McGregor's respective algorithms for MWM; the
novelty lies in the analysis for the MSM setting. In fact we identify a general
framework for MWM algorithms that allows this kind of adaptation to the broader
setting of MSM.
In the sequel, we give generalizations of these results where the
maximization is over "independent sets" in a very general sense. This
generalization captures hypermatchings in hypergraphs as well as independence
in the intersection of multiple matroids.Comment: 18 page
Streaming Verification of Graph Properties
Streaming interactive proofs (SIPs) are a framework for outsourced
computation. A computationally limited streaming client (the verifier) hands
over a large data set to an untrusted server (the prover) in the cloud and the
two parties run a protocol to confirm the correctness of result with high
probability. SIPs are particularly interesting for problems that are hard to
solve (or even approximate) well in a streaming setting. The most notable of
these problems is finding maximum matchings, which has received intense
interest in recent years but has strong lower bounds even for constant factor
approximations.
In this paper, we present efficient streaming interactive proofs that can
verify maximum matchings exactly. Our results cover all flavors of matchings
(bipartite/non-bipartite and weighted). In addition, we also present streaming
verifiers for approximate metric TSP. In particular, these are the first
efficient results for weighted matchings and for metric TSP in any streaming
verification model.Comment: 26 pages, 2 figure, 1 tabl
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]
Maximum Matching in Turnstile Streams
We consider the unweighted bipartite maximum matching problem in the one-pass
turnstile streaming model where the input stream consists of edge insertions
and deletions. In the insertion-only model, a one-pass -approximation
streaming algorithm can be easily obtained with space , where
denotes the number of vertices of the input graph. We show that no such result
is possible if edge deletions are allowed, even if space is
granted, for every . Specifically, for every , we show that in the one-pass turnstile streaming model, in order to compute
a -approximation, space is
required for constant error randomized algorithms, and, up to logarithmic
factors, space is sufficient. Our lower bound result is
proved in the simultaneous message model of communication and may be of
independent interest
Approximating Semi-Matchings in Streaming and in Two-Party Communication
We study the communication complexity and streaming complexity of
approximating unweighted semi-matchings. A semi-matching in a bipartite graph G
= (A, B, E), with n = |A|, is a subset of edges S that matches all A vertices
to B vertices with the goal usually being to do this as fairly as possible.
While the term 'semi-matching' was coined in 2003 by Harvey et al. [WADS 2003],
the problem had already previously been studied in the scheduling literature
under different names.
We present a deterministic one-pass streaming algorithm that for any 0 <=
\epsilon <= 1 uses space O(n^{1+\epsilon}) and computes an
O(n^{(1-\epsilon)/2})-approximation to the semi-matching problem. Furthermore,
with O(log n) passes it is possible to compute an O(log n)-approximation with
space O(n).
In the one-way two-party communication setting, we show that for every
\epsilon > 0, deterministic communication protocols for computing an
O(n^{1/((1+\epsilon)c + 1)})-approximation require a message of size more than
cn bits. We present two deterministic protocols communicating n and 2n edges
that compute an O(sqrt(n)) and an O(n^{1/3})-approximation respectively.
Finally, we improve on results of Harvey et al. [Journal of Algorithms 2006]
and prove new links between semi-matchings and matchings. While it was known
that an optimal semi-matching contains a maximum matching, we show that there
is a hierarchical decomposition of an optimal semi-matching into maximum
matchings. A similar result holds for semi-matchings that do not admit
length-two degree-minimizing paths.Comment: This is the long version including all proves of the ICALP 2013 pape
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