We consider the online bipartite matching problem on $(k,d)$-bounded graphs, where each online vertex has at most $d$ neighbors, each offline vertex has at least $k$ neighbors, and $k\geq d\geq 2$. The model of $(k,d)$-bounded graphs is proposed by Naor and Wajc (EC 2015 and TEAC 2018) to model the online advertising applications in which offline advertisers are interested in a large number of ad slots, while each online ad slot is interesting to a small number of advertisers. They proposed deterministic and randomized algorithms with a competitive ratio of $1 - (1-1/d)^k$ for the problem, and show that the competitive ratio is optimal for deterministic algorithms. They also raised the open questions of whether strictly better competitive ratios can be achieved using randomized algorithms, for both the adversarial and stochastic arrival models. In this paper we answer both of their open problems affirmatively. For the adversarial arrival model, we propose a randomized algorithm with competitive ratio $1 - (1-1/d)^k + \Omega(d^{-4}\cdot e^{-\frac{k}{d}})$ for all $k\geq d\geq 2$. We also consider the stochastic model and show that even better competitive ratios can be achieved. We show that for all $k\geq d\geq 2$, the competitive ratio is always at least $0.8237$. We further consider the $b$-matching problem when each offline vertex can be matched at most $b$ times, and provide several competitive ratio lower bounds for the adversarial and stochastic model.

翻译：我们考虑$(k,d)$-有界图上的在线二分匹配问题，其中每个在线顶点最多有$d$个邻居，每个离线顶点至少有$k$个邻居，且$k\geq d\geq 2$。$(k,d)$-有界图模型由Naor和Wajc（EC 2015和TEAC 2018）提出，用于建模在线广告应用场景，其中离线广告商对大量广告位感兴趣，而每个在线广告位仅对少数广告商有吸引力。他们针对该问题提出了竞争比为$1 - (1-1/d)^k$的确定性算法和随机算法，并证明了该竞争比对确定性算法是最优的。他们还提出了开放性问题：在对抗性和随机到达模型下，随机算法能否实现严格更优的竞争比。本文对这两个开放问题均给出肯定回答。对于对抗性到达模型，我们提出一种随机算法，对所有$k\geq d\geq 2$，其竞争比达到$1 - (1-1/d)^k + \Omega(d^{-4}\cdot e^{-\frac{k}{d}})$。我们还考虑随机模型，并证明可实现更优的竞争比：对所有$k\geq d\geq 2$，竞争比至少为$0.8237$。我们进一步研究每个离线顶点最多可匹配$b$次的$b$-匹配问题，并为对抗性和随机模型提供了若干竞争比下界。