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We introduce a new random input model for bipartite matching which we callthe Random Type Poisson Arrival Model. Just like in the known i.i.d. model(introduced by Feldman et al. 2009), online nodes have types in our model. Incontrast to the adversarial types studied in the known i.i.d. model, followingthe random graphs studied in Mastin and Jaillet 2016, in our model each typegraph is generated randomly by including each offline node in the neighborhoodof an online node with probability $c/n$ independently. In our model, nodes ofthe same type appear consecutively in the input and the number of times eachtype node appears is distributed according to the Poisson distribution withparameter 1. We analyze the performance of the simple greedy algorithm underthis input model. The performance is controlled by the parameter $c$ and we areable 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 ofthe matching in the remaining regime of constant $c$. We compare our results tothe previous work of Mastin and Jaillet who analyzed the simple greedyalgorithm in the $G_{n,n,p}$ model where each online node type occurs exactlyonce. We essentially show that the approach of Mastin and Jaillet can beextended to work for the Random Type Poisson Arrival Model, although severalnontrivial technical challenges need to be overcome. Intuitively, one can viewthe Random Type Poisson Arrival Model as the $G_{n,n,p}$ model with lessrandomness; that is, instead of each online node having a new type, each onlinenode has a chance of repeating the previous type.

Greedy Bipartite Matching in Random Type Poisson Arrival Model