686 research outputs found
Belief propagation : an asymptotically optimal algorithm for the random assignment problem
The random assignment problem asks for the minimum-cost perfect matching in
the complete bipartite graph \Knn with i.i.d. edge weights, say
uniform on . In a remarkable work by Aldous (2001), the optimal cost was
shown to converge to as , as conjectured by M\'ezard and
Parisi (1987) through the so-called cavity method. The latter also suggested a
non-rigorous decentralized strategy for finding the optimum, which turned out
to be an instance of the Belief Propagation (BP) heuristic discussed by Pearl
(1987). In this paper we use the objective method to analyze the performance of
BP as the size of the underlying graph becomes large. Specifically, we
establish that the dynamic of BP on \Knn converges in distribution as
to an appropriately defined dynamic on the Poisson Weighted
Infinite Tree, and we then prove correlation decay for this limiting dynamic.
As a consequence, we obtain that BP finds an asymptotically correct assignment
in time only. This contrasts with both the worst-case upper bound for
convergence of BP derived by Bayati, Shah and Sharma (2005) and the best-known
computational cost of achieved by Edmonds and Karp's algorithm
(1972).Comment: Mathematics of Operations Research (2009
Elastohydrodynamics of a sliding, spinning and sedimenting cylinder near a soft wall
We consider the motion of a fluid-immersed negatively buoyant particle in the
vicinity of a thin compressible elastic wall, a situation that arises in a
variety of technological and natural settings. We use scaling arguments to
establish different regimes of sliding, and complement these estimates using
thin-film lubrication dynamics to determine an asymptotic theory for the
sedimentation, sliding, and spinning motions of a cylinder. The resulting
theory takes the form of three coupled nonlinear singular-differential
equations. Numerical integration of the resulting equations confirms our
scaling relations and further yields a range of unexpected behaviours. Despite
the low-Reynolds feature of the flow, we demonstrate that the particle can
spontaneously oscillate when sliding, can generate lift via a Magnus-like
effect, can undergo a spin-induced reversal effect, and also shows an unusual
sedimentation singularity. Our description also allows us to address a
sedimentation-sliding transition that can lead to the particle coasting over
very long distances, similar to certain geophysical phenomena. Finally, we show
that a small modification of our theory allows to generalize the results to
account for additional effects such as wall poroelasticity
The densest subgraph problem in sparse random graphs
We determine the asymptotic behavior of the maximum subgraph density of large
random graphs with a prescribed degree sequence. The result applies in
particular to the Erd\H{o}s-R\'{e}nyi model, where it settles a conjecture of
Hajek [IEEE Trans. Inform. Theory 36 (1990) 1398-1414]. Our proof consists in
extending the notion of balanced loads from finite graphs to their local weak
limits, using unimodularity. This is a new illustration of the objective method
described by Aldous and Steele [In Probability on Discrete Structures (2004)
1-72 Springer].Comment: Published at http://dx.doi.org/10.1214/14-AAP1091 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
- …