Abstract
We introduce a new random input model for bipartite matching which we call the Random Type Poisson Arrival Model. Just like in the known i.i.d. model (introduced by Feldman et al. [Feldman et al., 2009]), online nodes have types in our model. In contrast to the adversarial types studied in the known i.i.d. model, following the random graphs studied in Mastin and Jaillet [A. Mastin, 2013], in our model each type graph is generated randomly by including each offline node in the neighborhood of an online node with probability c/n independently. In our model, nodes of the same type appear consecutively in the input and the number of times each type node appears is distributed according to the Poisson distribution with parameter 1. We analyze the performance of the simple greedy algorithm under this input model. The performance is controlled by the parameter c and we are able to exactly characterize the competitive ratio for the regimes c = o(1) and c = omega(1). We also provide a precise bound on the expected size of the matching in the remaining regime of constant c. We compare our results to the previous work of Mastin and Jaillet who analyzed the simple greedy algorithm in the G_{n,n,p} model where each online node type occurs exactly once. We essentially show that the approach of Mastin and Jaillet can be extended to work for the Random Type Poisson Arrival Model, although several nontrivial technical challenges need to be overcome. Intuitively, one can view the Random Type Poisson Arrival Model as the G_{n,n,p} model with less randomness; that is, instead of each online node having a new type, each online node has a chance of repeating the previous type.
BibTeX  Entry
@InProceedings{borodin_et_al:LIPIcs:2018:9409,
author = {Allan Borodin and Christodoulos Karavasilis and Denis Pankratov},
title = {{Greedy Bipartite Matching in Random Type Poisson Arrival Model}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {5:15:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770859},
ISSN = {18688969},
year = {2018},
volume = {116},
editor = {Eric Blais and Klaus Jansen and Jos{\'e} D. P. Rolim and David Steurer},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9409},
URN = {urn:nbn:de:0030drops94098},
doi = {10.4230/LIPIcs.APPROXRANDOM.2018.5},
annote = {Keywords: bipartite matching, stochastic input models, online algorithms, greedy algorithms}
}
Keywords: 

bipartite matching, stochastic input models, online algorithms, greedy algorithms 
Collection: 

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018) 
Issue Date: 

2018 
Date of publication: 

13.08.2018 