218,767 research outputs found
Strongly polynomial algorithm for a class of minimum-cost flow problems with separable convex objectives
A well-studied nonlinear extension of the minimum-cost flow problem is to
minimize the objective over feasible flows ,
where on every arc of the network, is a convex function. We give
a strongly polynomial algorithm for the case when all 's are convex
quadratic functions, settling an open problem raised e.g. by Hochbaum [1994].
We also give strongly polynomial algorithms for computing market equilibria in
Fisher markets with linear utilities and with spending constraint utilities,
that can be formulated in this framework (see Shmyrev [2009], Devanur et al.
[2011]). For the latter class this resolves an open question raised by Vazirani
[2010]. The running time is for quadratic costs,
for Fisher's markets with linear utilities and
for spending constraint utilities.
All these algorithms are presented in a common framework that addresses the
general problem setting. Whereas it is impossible to give a strongly polynomial
algorithm for the general problem even in an approximate sense (see Hochbaum
[1994]), we show that assuming the existence of certain black-box oracles, one
can give an algorithm using a strongly polynomial number of arithmetic
operations and oracle calls only. The particular algorithms can be derived by
implementing these oracles in the respective settings
Parallel Metric Tree Embedding based on an Algebraic View on Moore-Bellman-Ford
A \emph{metric tree embedding} of expected \emph{stretch~}
maps a weighted -node graph to a weighted tree with such that, for all ,
and
. Such embeddings are highly useful for designing
fast approximation algorithms, as many hard problems are easy to solve on tree
instances. However, to date the best parallel -depth algorithm that achieves an asymptotically optimal expected stretch of
requires
work and a metric as input.
In this paper, we show how to achieve the same guarantees using
depth and
work, where and is an arbitrarily small constant.
Moreover, one may further reduce the work to at the expense of increasing the expected stretch to
.
Our main tool in deriving these parallel algorithms is an algebraic
characterization of a generalization of the classic Moore-Bellman-Ford
algorithm. We consider this framework, which subsumes a variety of previous
"Moore-Bellman-Ford-like" algorithms, to be of independent interest and discuss
it in depth. In our tree embedding algorithm, we leverage it for providing
efficient query access to an approximate metric that allows sampling the tree
using depth and work.
We illustrate the generality and versatility of our techniques by various
examples and a number of additional results
Faster Algorithms for Semi-Matching Problems
We consider the problem of finding \textit{semi-matching} in bipartite graphs
which is also extensively studied under various names in the scheduling
literature. We give faster algorithms for both weighted and unweighted case.
For the weighted case, we give an -time algorithm, where is
the number of vertices and is the number of edges, by exploiting the
geometric structure of the problem. This improves the classical
algorithms by Horn [Operations Research 1973] and Bruno, Coffman and Sethi
[Communications of the ACM 1974].
For the unweighted case, the bound could be improved even further. We give a
simple divide-and-conquer algorithm which runs in time,
improving two previous -time algorithms by Abraham [MSc thesis,
University of Glasgow 2003] and Harvey, Ladner, Lov\'asz and Tamir [WADS 2003
and Journal of Algorithms 2006]. We also extend this algorithm to solve the
\textit{Balance Edge Cover} problem in time, improving the
previous -time algorithm by Harada, Ono, Sadakane and Yamashita [ISAAC
2008].Comment: ICALP 201
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