Greedy Bipartite Matching in Random Type Poisson Arrival Model

TL;DR

This paper introduces the Random Type Poisson Arrival Model for bipartite matching, analyzes the greedy algorithm's performance across different regimes of the parameter c, and extends previous models to a less random setting.

Contribution

The paper defines a new bipartite matching input model, analyzes the greedy algorithm's performance in this model, and extends prior work from the G_{n,n,p} model to this new setting.

Findings

Exact competitive ratio for c = o(1) and c = ω(1) regimes.

Precise bounds on expected matching size for constant c.

Extension of Mastin and Jaillet's approach to the new model.

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. 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 2016, 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.