Scalable Auction Algorithms for Bipartite Maximum Matching Problems

Abstract

In this paper, we give new auction algorithms for maximum weighted bipartite matching (MWM) and maximum cardinality bipartite $b$-matching (MCbM). Our algorithms run in $O\left(\log n/\varepsilon^8\right)$ and $O\left(\log n/\varepsilon^2\right)$ rounds, respectively, in the blackboard distributed setting. We show that our MWM algorithm can be implemented in the distributed, interactive setting using $O(\log^2 n)$ and $O(\log n)$ bit messages, respectively, directly answering the open question posed by Demange, Gale and Sotomayor [DNO14]. Furthermore, we implement our algorithms in a variety of other models including the the semi-streaming model, the shared-memory work-depth model, and the massively parallel computation model. Our semi-streaming MWM algorithm uses $O(1/\varepsilon^8)$ passes in $O(n \log n \cdot \log(1/\varepsilon))$ space and our MCbM algorithm runs in $O(1/\varepsilon^2)$ passes using $O\left(\left(\sum_{i \in L} b_i + |R|\right)\log(1/\varepsilon)\right)$ space (where parameters $b_i$ represent the degree constraints on the $b$-matching and $L$ and $R$ represent the left and right side of the bipartite graph, respectively). Both of these algorithms improves \emph{exponentially} the dependence on $\varepsilon$ in the space complexity in the semi-streaming model against the best-known algorithms for these problems, in addition to improvements in round complexity for MCbM. Finally, our algorithms eliminate the large polylogarithmic dependence on $n$ in depth and number of rounds in the work-depth and massively parallel computation models, respectively, improving on previous results which have large polylogarithmic dependence on $n$ (and exponential dependence on $\varepsilon$ in the MPC model).