On the communication and streaming complexity of maximum bipartite matching

• Authors: Ashish Goel;Michael Kapralov;Sanjeev Khanna
• Venue: Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
• Year: 2012

Consider the following communication problem. Alice holds a graph GA = (P,Q,EA) and Bob holds a graph GB = (P,Q,EB), where |P| = |Q| = n. Alice is allowed to send Bob a message m that depends only on the graph GA. Bob must then output a matching M ⊆ EA ∪ EB. What is the minimum message size of the message m that Alice sends to Bob that allows Bob to recover a matching of size at least (1 − ε) times the maximum matching in GA ∪ GB? The minimum message length is the one-round communication complexity of approximating bipartite matching. It is easy to see that the one-round communication complexity also gives a lower bound on the space needed by a one-pass streaming algorithm to compute a (1 − ε)-approximate bipartite matching. The focus of this work is to understand one-round communication complexity and one-pass streaming complexity of maximum bipartite matching. In particular, how well can one approximate these problems with linear communication and space? Prior to our work, only a 1/2-approximation was known for both these problems. In order to study these questions, we introduce the concept of an ε-matching cover of a bipartite graph G, which is a sparse subgraph of the original graph that preserves the size of maximum matching between every subset of vertices to within an additive εn error. We give a polynomial time construction of a 1/2-matching cover of size O(n) with some crucial additional properties, thereby showing that Alice and Bob can achieve a 2/3-approximation with a message of size O(n). While we do not provide bounds on the size of ε-matching covers for ε G on n vertices is essentially equal to the size of the largest so-called ε-Ruzsa Szemerédi graph on n vertices. We use this connection to show that for any δ 0, a (2/3 + δ)-approximation requires a communication complexity of n1+Ω(1/log log n). We also consider the natural restrictingon of the problem in which GA and GB are only allowed to share vertices on one side of the bipartition, which is motivated by applications to one-pass streaming with vertex arrivals. We show that a 3/4-approximation can be achieved with a linear size message in this case, and this result is best possible in that super-linear space is needed to achieve any better approximation. Finally, we build on our techniques for the restricted version above to design one-pass streaming algorithm for the case when vertices on one side are known in advance, and the vertices on the other side arrive in a streaming manner together with all their incident edges. This is precisely the setting of the celebrated (1 − 1/ε)-competitive randomized algorithm of Karp-Vazirani-Vazirani (KVV) for the online bipartite matching problem [12]. We present here the first deterministic one-pass streaming (1 - 1/ε)-approximation algorithm using O(n) space for this setting.