79,907 research outputs found
Minimum-cost matching in a random graph with random costs
Let be the standard Erd\H{o}s-R\'enyi-Gilbert random graph and let
be the random bipartite graph on vertices, where each appears as an edge independently with probability . For a graph
, suppose that each edge is given an independent uniform
exponential rate one cost. Let denote the random variable equal to the
length of the minimum cost perfect matching, assuming that contains at
least one. We show that w.h.p. if then w.h.p. {\bf
E}[C(G_{n,n,p})] =(1+o(1))\frac{\p^2}{6p}. This generalises the well-known
result for the case . We also show that w.h.p. {\bf E}[C(G_{n,p})]
=(1+o(1))\frac{\p^2}{12p} along with concentration results for both types of
random graph.Comment: Replaces an earlier paper where was an arbitrary regular
bipartite grap
The Price of Information in Combinatorial Optimization
Consider a network design application where we wish to lay down a
minimum-cost spanning tree in a given graph; however, we only have stochastic
information about the edge costs. To learn the precise cost of any edge, we
have to conduct a study that incurs a price. Our goal is to find a spanning
tree while minimizing the disutility, which is the sum of the tree cost and the
total price that we spend on the studies. In a different application, each edge
gives a stochastic reward value. Our goal is to find a spanning tree while
maximizing the utility, which is the tree reward minus the prices that we pay.
Situations such as the above two often arise in practice where we wish to
find a good solution to an optimization problem, but we start with only some
partial knowledge about the parameters of the problem. The missing information
can be found only after paying a probing price, which we call the price of
information. What strategy should we adopt to optimize our expected
utility/disutility?
A classical example of the above setting is Weitzman's "Pandora's box"
problem where we are given probability distributions on values of
independent random variables. The goal is to choose a single variable with a
large value, but we can find the actual outcomes only after paying a price. Our
work is a generalization of this model to other combinatorial optimization
problems such as matching, set cover, facility location, and prize-collecting
Steiner tree. We give a technique that reduces such problems to their non-price
counterparts, and use it to design exact/approximation algorithms to optimize
our utility/disutility. Our techniques extend to situations where there are
additional constraints on what parameters can be probed or when we can
simultaneously probe a subset of the parameters.Comment: SODA 201
Belief propagation for optimal edge cover in the random complete graph
We apply the objective method of Aldous to the problem of finding the
minimum-cost edge cover of the complete graph with random independent and
identically distributed edge costs. The limit, as the number of vertices goes
to infinity, of the expected minimum cost for this problem is known via a
combinatorial approach of Hessler and W\"{a}stlund. We provide a proof of this
result using the machinery of the objective method and local weak convergence,
which was used to prove the limit of the random assignment problem.
A proof via the objective method is useful because it provides us with more
information on the nature of the edge's incident on a typical root in the
minimum-cost edge cover. We further show that a belief propagation algorithm
converges asymptotically to the optimal solution. This can be applied in a
computational linguistics problem of semantic projection. The belief
propagation algorithm yields a near optimal solution with lesser complexity
than the known best algorithms designed for optimality in worst-case settings.Comment: Published in at http://dx.doi.org/10.1214/13-AAP981 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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