1,332 research outputs found
Robust randomized matchings
The following game is played on a weighted graph: Alice selects a matching
and Bob selects a number . Alice's payoff is the ratio of the weight of
the heaviest edges of to the maximum weight of a matching of size at
most . If guarantees a payoff of at least then it is called
-robust. In 2002, Hassin and Rubinstein gave an algorithm that returns
a -robust matching, which is best possible.
We show that Alice can improve her payoff to by playing a
randomized strategy. This result extends to a very general class of
independence systems that includes matroid intersection, b-matchings, and
strong 2-exchange systems. It also implies an improved approximation factor for
a stochastic optimization variant known as the maximum priority matching
problem and translates to an asymptotic robustness guarantee for deterministic
matchings, in which Bob can only select numbers larger than a given constant.
Moreover, we give a new LP-based proof of Hassin and Rubinstein's bound
Matchings in Random Biregular Bipartite Graphs
We study the existence of perfect matchings in suitably chosen induced
subgraphs of random biregular bipartite graphs. We prove a result similar to a
classical theorem of Erdos and Renyi about perfect matchings in random
bipartite graphs. We also present an application to commutative graphs, a class
of graphs that are featured in additive number theory.Comment: 30 pages and 3 figures - Latest version has updated introduction and
bibliograph
Matchings on infinite graphs
Elek and Lippner (2010) showed that the convergence of a sequence of
bounded-degree graphs implies the existence of a limit for the proportion of
vertices covered by a maximum matching. We provide a characterization of the
limiting parameter via a local recursion defined directly on the limit of the
graph sequence. Interestingly, the recursion may admit multiple solutions,
implying non-trivial long-range dependencies between the covered vertices. We
overcome this lack of correlation decay by introducing a perturbative parameter
(temperature), which we let progressively go to zero. This allows us to
uniquely identify the correct solution. In the important case where the graph
limit is a unimodular Galton-Watson tree, the recursion simplifies into a
distributional equation that can be solved explicitly, leading to a new
asymptotic formula that considerably extends the well-known one by Karp and
Sipser for Erd\"os-R\'enyi random graphs.Comment: 23 page
Percolation on sparse random graphs with given degree sequence
We study the two most common types of percolation process on a sparse random
graph with a given degree sequence. Namely, we examine first a bond percolation
process where the edges of the graph are retained with probability p and
afterwards we focus on site percolation where the vertices are retained with
probability p. We establish critical values for p above which a giant component
emerges in both cases. Moreover, we show that in fact these coincide. As a
special case, our results apply to power law random graphs. We obtain rigorous
proofs for formulas derived by several physicists for such graphs.Comment: 20 page
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