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August 2017 Phase transition in a sequential assignment problem on graphs
Antal A. Járai
Ann. Appl. Probab. 27(4): 2098-2129 (August 2017). DOI: 10.1214/16-AAP1250

Abstract

We study the following sequential assignment problem on a finite graph G=(V,E). Each edge eE starts with an integer value ne0, and we write n=eEne. At time t, 1tn, a uniformly random vertex vV is generated, and one of the edges f incident with v must be selected. The value of f is then decreased by 1. There is a unit final reward if the configuration (0,,0) is reached. Our main result is that there is a phase transition: as n, the expected reward under the optimal policy approaches a constant cG>0 when (ne/n:eE) converges to a point in the interior of a certain convex set RG, and goes to 0 exponentially when (ne/n:eE) is bounded away from RG. We also obtain estimates in the near-critical region, that is when (ne/n:eE) lies close to RG. We supply quantitative error bounds in our arguments.

Citation

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Antal A. Járai. "Phase transition in a sequential assignment problem on graphs." Ann. Appl. Probab. 27 (4) 2098 - 2129, August 2017. https://doi.org/10.1214/16-AAP1250

Information

Received: 1 July 2015; Revised: 1 September 2016; Published: August 2017
First available in Project Euclid: 30 August 2017

zbMATH: 06803458
MathSciNet: MR3693521
Digital Object Identifier: 10.1214/16-AAP1250

Subjects:
Primary: 60K99
Secondary: 90C40 , 91A60

Keywords: critical phenomenon , discrete stochastic optimal control , Markov decision process , phase transition , stochastic dynamic programming , stochastic sequential assignment

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.27 • No. 4 • August 2017
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