On the communication and streaming complexity of maximum bipartite matching [chapter]

Ashish Goel, Michael Kapralov, Sanjeev Khanna
2012 Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms  
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 − ) times the maximum matching in GA ∪ GB? The minimum message length is the one-round communication
more » ... plexity 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 − )-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 -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 n 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 -matching covers for < 1/2, we prove that in general, the size of the smallest -matching cover of a graph G on n vertices is essentially equal to the size of the largest so-called -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 n 1+Ω(1/ log log n) .
doi:10.1137/1.9781611973099.41 dblp:conf/soda/GoelKK12 fatcat:gzjojswvznbche36dlffke5rzi