30,447 research outputs found
New deterministic approximation algorithms for fully dynamic matching
We present two deterministic dynamic algorithms for the maximum matching
problem. (1) An algorithm that maintains a -approximate maximum
matching in general graphs with update
time. (2) An algorithm that maintains an approximation of the {\em
value} of the maximum matching with update time in bipartite
graphs, for every sufficiently large constant positive integer . Here,
is a constant determined by the value of . Result (1)
is the first deterministic algorithm that can maintain an -approximate maximum matching with polylogarithmic update time, improving
the seminal result of Onak et al. [STOC 2010]. Its approximation guarantee
almost matches the guarantee of the best {\em randomized} polylogarithmic
update time algorithm [Baswana et al. FOCS 2011]. Result (2) achieves a
better-than-two approximation with {\em arbitrarily small polynomial} update
time on bipartite graphs. Previously the best update time for this problem was
[Bernstein et al. ICALP 2015], where is the current number of
edges in the graph.Comment: To appear in STOC 201
On Regularity Lemma and Barriers in Streaming and Dynamic Matching
We present a new approach for finding matchings in dense graphs by building
on Szemer\'edi's celebrated Regularity Lemma. This allows us to obtain
non-trivial albeit slight improvements over longstanding bounds for matchings
in streaming and dynamic graphs. In particular, we establish the following
results for -vertex graphs:
* A deterministic single-pass streaming algorithm that finds
a -approximate matching in bits of space. This constitutes
the first single-pass algorithm for this problem in sublinear space that
improves over the -approximation of the greedy algorithm.
* A randomized fully dynamic algorithm that with high probability maintains a
-approximate matching in worst-case update time per each edge
insertion or deletion. The algorithm works even against an adaptive adversary.
This is the first update-time dynamic algorithm with approximation
guarantee arbitrarily close to one.
Given the use of regularity lemma, the improvement obtained by our algorithms
over trivial bounds is only by some factor.
Nevertheless, in each case, they show that the ``right'' answer to the problem
is not what is dictated by the previous bounds.
Finally, in the streaming model, we also present a randomized
-approximation algorithm whose space can be upper bounded by the
density of certain Ruzsa-Szemer\'edi (RS) graphs. While RS graphs by now have
been used extensively to prove streaming lower bounds, ours is the first to use
them as an upper bound tool for designing improved streaming algorithms
Fully Dynamic Matching in Bipartite Graphs
Maximum cardinality matching in bipartite graphs is an important and
well-studied problem. The fully dynamic version, in which edges are inserted
and deleted over time has also been the subject of much attention. Existing
algorithms for dynamic matching (in general graphs) seem to fall into two
groups: there are fast (mostly randomized) algorithms that do not achieve a
better than 2-approximation, and there slow algorithms with \O(\sqrt{m})
update time that achieve a better-than-2 approximation. Thus the obvious
question is whether we can design an algorithm -- deterministic or randomized
-- that achieves a tradeoff between these two: a approximation
and a better-than-2 approximation simultaneously. We answer this question in
the affirmative for bipartite graphs.
Our main result is a fully dynamic algorithm that maintains a 3/2 + \eps
approximation in worst-case update time O(m^{1/4}\eps^{/2.5}). We also give
stronger results for graphs whose arboricity is at most \al, achieving a (1+
\eps) approximation in worst-case time O(\al (\al + \log n)) for constant
\eps. When the arboricity is constant, this bound is and when the
arboricity is polylogarithmic the update time is also polylogarithmic.
The most important technical developement is the use of an intermediate graph
we call an edge degree constrained subgraph (EDCS). This graph places
constraints on the sum of the degrees of the endpoints of each edge: upper
bounds for matched edges and lower bounds for unmatched edges. The main
technical content of our paper involves showing both how to maintain an EDCS
dynamically and that and EDCS always contains a sufficiently large matching. We
also make use of graph orientations to help bound the amount of work done
during each update.Comment: Longer version of paper that appears in ICALP 201
Distributed PCP Theorems for Hardness of Approximation in P
We present a new distributed model of probabilistically checkable proofs
(PCP). A satisfying assignment to a CNF formula is
shared between two parties, where Alice knows , Bob knows
, and both parties know . The goal is to have
Alice and Bob jointly write a PCP that satisfies , while
exchanging little or no information. Unfortunately, this model as-is does not
allow for nontrivial query complexity. Instead, we focus on a non-deterministic
variant, where the players are helped by Merlin, a third party who knows all of
.
Using our framework, we obtain, for the first time, PCP-like reductions from
the Strong Exponential Time Hypothesis (SETH) to approximation problems in P.
In particular, under SETH we show that there are no truly-subquadratic
approximation algorithms for Bichromatic Maximum Inner Product over
{0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate
Regular Expression Matching, and Diameter in Product Metric. All our
inapproximability factors are nearly-tight. In particular, for the first two
problems we obtain nearly-polynomial factors of ; only
-factor lower bounds (under SETH) were known before
Optimal Dynamic Distributed MIS
Finding a maximal independent set (MIS) in a graph is a cornerstone task in
distributed computing. The local nature of an MIS allows for fast solutions in
a static distributed setting, which are logarithmic in the number of nodes or
in their degrees. The result trivially applies for the dynamic distributed
model, in which edges or nodes may be inserted or deleted. In this paper, we
take a different approach which exploits locality to the extreme, and show how
to update an MIS in a dynamic distributed setting, either \emph{synchronous} or
\emph{asynchronous}, with only \emph{a single adjustment} and in a single
round, in expectation. These strong guarantees hold for the \emph{complete
fully dynamic} setting: Insertions and deletions, of edges as well as nodes,
gracefully and abruptly. This strongly separates the static and dynamic
distributed models, as super-constant lower bounds exist for computing an MIS
in the former.
Our results are obtained by a novel analysis of the surprisingly simple
solution of carefully simulating the greedy \emph{sequential} MIS algorithm
with a random ordering of the nodes. As such, our algorithm has a direct
application as a -approximation algorithm for correlation clustering. This
adds to the important toolbox of distributed graph decompositions, which are
widely used as crucial building blocks in distributed computing.
Finally, our algorithm enjoys a useful \emph{history-independence} property,
meaning the output is independent of the history of topology changes that
constructed that graph. This means the output cannot be chosen, or even biased,
by the adversary in case its goal is to prevent us from optimizing some
objective function.Comment: 19 pages including appendix and reference
On Conceptually Simple Algorithms for Variants of Online Bipartite Matching
We present a series of results regarding conceptually simple algorithms for
bipartite matching in various online and related models. We first consider a
deterministic adversarial model. The best approximation ratio possible for a
one-pass deterministic online algorithm is , which is achieved by any
greedy algorithm. D\"urr et al. recently presented a -pass algorithm called
Category-Advice that achieves approximation ratio . We extend their
algorithm to multiple passes. We prove the exact approximation ratio for the
-pass Category-Advice algorithm for all , and show that the
approximation ratio converges to the inverse of the golden ratio
as goes to infinity. The convergence is
extremely fast --- the -pass Category-Advice algorithm is already within
of the inverse of the golden ratio.
We then consider a natural greedy algorithm in the online stochastic IID
model---MinDegree. This algorithm is an online version of a well-known and
extensively studied offline algorithm MinGreedy. We show that MinDegree cannot
achieve an approximation ratio better than , which is guaranteed by any
consistent greedy algorithm in the known IID model.
Finally, following the work in Besser and Poloczek, we depart from an
adversarial or stochastic ordering and investigate a natural randomized
algorithm (MinRanking) in the priority model. Although the priority model
allows the algorithm to choose the input ordering in a general but well defined
way, this natural algorithm cannot obtain the approximation of the Ranking
algorithm in the ROM model
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