254,876 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
Efficient Algorithms for Scheduling Moldable Tasks
We study the problem of scheduling independent moldable tasks on
processors that arises in large-scale parallel computations. When tasks are
monotonic, the best known result is a -approximation
algorithm for makespan minimization with a complexity linear in and
polynomial in and where is
arbitrarily small. We propose a new perspective of the existing speedup models:
the speedup of a task is linear when the number of assigned
processors is small (up to a threshold ) while it presents
monotonicity when ranges in ; the bound
indicates an unacceptable overhead when parallelizing on too many processors.
For a given integer , let . In this paper, we propose a -approximation algorithm for makespan minimization with a
complexity where
(). As
a by-product, we also propose a -approximation algorithm for
throughput maximization with a common deadline with a complexity
Distributed Approximation Algorithms for Weighted Shortest Paths
A distributed network is modeled by a graph having nodes (processors) and
diameter . We study the time complexity of approximating {\em weighted}
(undirected) shortest paths on distributed networks with a {\em
bandwidth restriction} on edges (the standard synchronous \congest model). The
question whether approximation algorithms help speed up the shortest paths
(more precisely distance computation) was raised since at least 2004 by Elkin
(SIGACT News 2004). The unweighted case of this problem is well-understood
while its weighted counterpart is fundamental problem in the area of
distributed approximation algorithms and remains widely open. We present new
algorithms for computing both single-source shortest paths (\sssp) and
all-pairs shortest paths (\apsp) in the weighted case.
Our main result is an algorithm for \sssp. Previous results are the classic
-time Bellman-Ford algorithm and an -time
-approximation algorithm, for any integer
, which follows from the result of Lenzen and Patt-Shamir (STOC 2013).
(Note that Lenzen and Patt-Shamir in fact solve a harder problem, and we use
to hide the O(\poly\log n) term.) We present an -time -approximation algorithm for \sssp. This
algorithm is {\em sublinear-time} as long as is sublinear, thus yielding a
sublinear-time algorithm with almost optimal solution. When is small, our
running time matches the lower bound of by Das Sarma
et al. (SICOMP 2012), which holds even when , up to a
\poly\log n factor.Comment: Full version of STOC 201
Algorithms for āp Low Rank Approximation
We consider the problem of approximating a given matrix by a low-rank matrix so as to minimize the entry-wise āp-approximation error, for any P ā„ 1; the case p = 2 is the classical SVD problem. We obtain the first provably good approximation algorithms for this version of low-rank approximation that work for every value of p ā„ 1, including p = Ļ. Our algorithms are simple, easy to implement, work well in practice, and illustrate interesting tradeoffs between the approximation quality, the running time, and the rank of the approximating matrix
Towards a better approximation for sparsest cut?
We give a new -approximation for sparsest cut problem on graphs
where small sets expand significantly more than the sparsest cut (sets of size
expand by a factor bigger, for some small ; this
condition holds for many natural graph families). We give two different
algorithms. One involves Guruswami-Sinop rounding on the level- Lasserre
relaxation. The other is combinatorial and involves a new notion called {\em
Small Set Expander Flows} (inspired by the {\em expander flows} of ARV) which
we show exists in the input graph. Both algorithms run in time . We also show similar approximation algorithms in graphs with
genus with an analogous local expansion condition. This is the first
algorithm we know of that achieves -approximation on such general
family of graphs
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