Scalable Auction Algorithms for Bipartite Maximum Matching Problems

TL;DR

This paper introduces new scalable auction algorithms for bipartite maximum matching problems, achieving significant improvements in round complexity, message efficiency, and space complexity across multiple computational models.

Contribution

The paper presents novel auction algorithms for bipartite maximum matching that are faster, more space-efficient, and applicable in various distributed and streaming models, answering open questions.

Findings

Algorithms run in logarithmic rounds with respect to n.

Semi-streaming algorithms use exponentially less space in epsilon dependence.

Improvements over previous methods in round complexity and space usage.

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).