Article

Bivariate Uniqueness and Endogeny for the Logistic Recursive Distributional Equation

02/2004;
Source: arXiv

ABSTRACT In this article we prove the bivariate uniqueness property for a particular ``max-type'' recursive distributional equation (RDE). Using the general theory developed by Aldous and Bandyopadhyay (2005) we then show that the corresponding recursive tree process (RTP) has no external randomness, more preciously, the RTP is endogenous. The RDE we consider is so called the Logistic RDE, which appears in Aldous' (2001) proof of the $\zeta(2)$-limit of the random assignment problem using the local weak convergence method. Thus this work provides a non-trivial application of the general theory developed by Aldous and Bandyopadhyay (2005).

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    ABSTRACT: 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\"astlund. We provide a proof of this result using the machinery of the objective method and local weak convergence, which was used to prove the \zeta(2) limit of the random assignment problem. A proof via the objective method is useful because it provides us more information on the nature of the edges 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 finds application 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.
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