75 research outputs found
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
On the Power of Advice and Randomization for Online Bipartite Matching
While randomized online algorithms have access to a sequence of uniform
random bits, deterministic online algorithms with advice have access to a
sequence of advice bits, i.e., bits that are set by an all powerful oracle
prior to the processing of the request sequence. Advice bits are at least as
helpful as random bits, but how helpful are they? In this work, we investigate
the power of advice bits and random bits for online maximum bipartite matching
(MBM).
The well-known Karp-Vazirani-Vazirani algorithm is an optimal randomized
-competitive algorithm for \textsc{MBM} that requires access
to uniform random bits. We show that
advice bits are necessary and
sufficient in order to obtain a
-competitive deterministic advice algorithm. Furthermore, for a
large natural class of deterministic advice algorithms, we prove that
advice bits are required in order to improve on the
-competitiveness of the best deterministic online algorithm, while
it is known that bits are sufficient.
Last, we give a randomized online algorithm that uses random bits, for
integers , and a competitive ratio that approaches
very quickly as is increasing. For example if , then the difference
between and the achieved competitive ratio is less than
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
Sublinear Estimation of Weighted Matchings in Dynamic Data Streams
This paper presents an algorithm for estimating the weight of a maximum
weighted matching by augmenting any estimation routine for the size of an
unweighted matching. The algorithm is implementable in any streaming model
including dynamic graph streams. We also give the first constant estimation for
the maximum matching size in a dynamic graph stream for planar graphs (or any
graph with bounded arboricity) using space which also
extends to weighted matching. Using previous results by Kapralov, Khanna, and
Sudan (2014) we obtain a approximation for general graphs
using space in random order streams, respectively. In
addition, we give a space lower bound of for any
randomized algorithm estimating the size of a maximum matching up to a
factor for adversarial streams
Semi-Streaming Set Cover
This paper studies the set cover problem under the semi-streaming model. The
underlying set system is formalized in terms of a hypergraph whose
edges arrive one-by-one and the goal is to construct an edge cover with the objective of minimizing the cardinality (or cost in the weighted
case) of . We consider a parameterized relaxation of this problem, where
given some , the goal is to construct an edge -cover, namely, a subset of edges incident to all but an
-fraction of the vertices (or their benefit in the weighted case).
The key limitation imposed on the algorithm is that its space is limited to
(poly)logarithmically many bits per vertex.
Our main result is an asymptotically tight trade-off between and
the approximation ratio: We design a semi-streaming algorithm that on input
graph , constructs a succinct data structure such that for
every , an edge -cover that approximates
the optimal edge \mbox{(-)cover} within a factor of can be
extracted from (efficiently and with no additional space
requirements), where In particular for the traditional
set cover problem we obtain an -approximation. This algorithm is
proved to be best possible by establishing a family (parameterized by
) of matching lower bounds.Comment: Full version of the extended abstract that will appear in Proceedings
of ICALP 2014 track
Almost Optimal Streaming Algorithms for Coverage Problems
Maximum coverage and minimum set cover problems --collectively called
coverage problems-- have been studied extensively in streaming models. However,
previous research not only achieve sub-optimal approximation factors and space
complexities, but also study a restricted set arrival model which makes an
explicit or implicit assumption on oracle access to the sets, ignoring the
complexity of reading and storing the whole set at once. In this paper, we
address the above shortcomings, and present algorithms with improved
approximation factor and improved space complexity, and prove that our results
are almost tight. Moreover, unlike most of previous work, our results hold on a
more general edge arrival model. More specifically, we present (almost) optimal
approximation algorithms for maximum coverage and minimum set cover problems in
the streaming model with an (almost) optimal space complexity of
, i.e., the space is {\em independent of the size of the sets or
the size of the ground set of elements}. These results not only improve over
the best known algorithms for the set arrival model, but also are the first
such algorithms for the more powerful {\em edge arrival} model. In order to
achieve the above results, we introduce a new general sketching technique for
coverage functions: This sketching scheme can be applied to convert an
-approximation algorithm for a coverage problem to a
(1-\eps)\alpha-approximation algorithm for the same problem in streaming, or
RAM models. We show the significance of our sketching technique by ruling out
the possibility of solving coverage problems via accessing (as a black box) a
(1 \pm \eps)-approximate oracle (e.g., a sketch function) that estimates the
coverage function on any subfamily of the sets
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