Markovian Online Matching Algorithms on Large Bipartite Random Graphs

Mohamed Habib Aliou Diallo Aoudi, Pascal Moyal () and Vincent Robin
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Mohamed Habib Aliou Diallo Aoudi: Alliance Sorbonne Universités, LMAC, Université de Technologie de Compiègne
Pascal Moyal: Université de Lorraine and INRIA / PASTA
Vincent Robin: Alliance Sorbonne Universités, LMAC, Université de Technologie de Compiègne

Methodology and Computing in Applied Probability, 2022, vol. 24, issue 4, 3195-3225

Abstract: Abstract In this paper, we present an approximation of the matching coverage on large bipartite graphs, for local online matching algorithms based on the sole knowledge of the remaining degree of the nodes of the graph at hand. This approximation is obtained by applying the Differential Equation Method to a measure-valued process representing an alternative construction, in which the matching and the graph are constructed simultaneously, by a uniform pairing leading to a realization of the bipartite Configuration Model. The latter auxiliary construction is shown to be equivalent in distribution to the original one. It allows to drastically reduce the complexity of the problem, in that the resulting matching coverage can be written as a simple function of the final value of the process, and in turn, approximated by a simple function of the solution of a system of ODE’s. By way of simulations, we illustrate the accuracy of our estimate, and then compare the performance of an algorithm based on the minimal residual degree of the nodes, to the classical greedy matching.

Keywords: Random graphs; Matching; Online algorithms; Stochastic processes (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s11009-022-09973-y

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