79,907 research outputs found

    Minimum-cost matching in a random graph with random costs

    Full text link
    Let Gn,pG_{n,p} be the standard Erd\H{o}s-R\'enyi-Gilbert random graph and let Gn,n,pG_{n,n,p} be the random bipartite graph on n+nn+n vertices, where each e[n]2e\in [n]^2 appears as an edge independently with probability pp. For a graph G=(V,E)G=(V,E), suppose that each edge eEe\in E is given an independent uniform exponential rate one cost. Let C(G)C(G) denote the random variable equal to the length of the minimum cost perfect matching, assuming that GG contains at least one. We show that w.h.p. if d=np(logn)2d=np\gg(\log n)^2 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 G=Kn,nG=K_{n,n}. 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 GG was an arbitrary regular bipartite grap

    The Price of Information in Combinatorial Optimization

    Full text link
    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 nn 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

    Full text link
    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 ζ(2)\zeta(2) 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
    corecore