55,658 research outputs found
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
Hierarchy construction schemes within the Scale set framework
Segmentation algorithms based on an energy minimisation framework often
depend on a scale parameter which balances a fit to data and a regularising
term. Irregular pyramids are defined as a stack of graphs successively reduced.
Within this framework, the scale is often defined implicitly as the height in
the pyramid. However, each level of an irregular pyramid can not usually be
readily associated to the global optimum of an energy or a global criterion on
the base level graph. This last drawback is addressed by the scale set
framework designed by Guigues. The methods designed by this author allow to
build a hierarchy and to design cuts within this hierarchy which globally
minimise an energy. This paper studies the influence of the construction scheme
of the initial hierarchy on the resulting optimal cuts. We propose one
sequential and one parallel method with two variations within both. Our
sequential methods provide partitions near the global optima while parallel
methods require less execution times than the sequential method of Guigues even
on sequential machines
Welfare Maximization with Limited Interaction
We continue the study of welfare maximization in unit-demand (matching)
markets, in a distributed information model where agent's valuations are
unknown to the central planner, and therefore communication is required to
determine an efficient allocation. Dobzinski, Nisan and Oren (STOC'14) showed
that if the market size is , then rounds of interaction (with
logarithmic bandwidth) suffice to obtain an -approximation to the
optimal social welfare. In particular, this implies that such markets converge
to a stable state (constant approximation) in time logarithmic in the market
size.
We obtain the first multi-round lower bound for this setup. We show that even
if the allowable per-round bandwidth of each agent is , the
approximation ratio of any -round (randomized) protocol is no better than
, implying an lower bound on the
rate of convergence of the market to equilibrium.
Our construction and technique may be of interest to round-communication
tradeoffs in the more general setting of combinatorial auctions, for which the
only known lower bound is for simultaneous () protocols [DNO14]
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