Bipartite Graph Matching in the Semi-Streaming Model

We give an introduction to classical algorithms for the maximum matching problem in bipartite graphs and then present a 1` ε approximation algorithm for the same problem in bipartite graph streams, which only requires Opn ̈ poly lognq bits of random-access memory and Opε ́5q passes over the input. All the algorithms are combinatorial in nature and use the augmenting paths technique. The streaming part of these notes is based on joint work with Sebastian Eggert, Peter Munstermann, and Anand Srivastav from Kiel University [EKM+12]. Basic Conventions. We use N “ {1, 2, 3, . . .} and N0 “ {0, 1, 2, 3, . . .}. We use Op ̈q and op ̈q to indicate an upper bound, making no statement about a lower bound. We use Ωp ̈q to indicate a lower bound, making no statement about an upper bound. 1 Graphs and Matching A graph is a pair G “ pV,Eq where V is a finite set and E Ď ( V 2 ) “ { {u, v} ; u, v P V ^ u ‰ v } . The elements of V are called vertices a.k.a. nodes and those of E are called edgs a.k.a. links. Let G “ pV,Eq be a graph. A sequence of vertices S “ pv0, . . . , vkq is called a walk of length k if {vi, vi`1} P E for all 0 ď i ă k. The vertices v0 and vk are the end-vertices of the walk, and the walk is also called a v0-vk walk. The walk is called a path if all the vertices are distinct, i. e., if |{v0, . . . , vk}| “ k ` 1. The walk is called a cycle if k ě 3 and pv0, . . . , vk ́1q is a path and v0 “ vk. If S is a walk, denote V pSq “ {v0, . . . , vk} the vertices visited by S and EpSq “ {{vi, vi`1} ; 0 ď i ă k} the edges traversed by S. Sometimes we write a walk in a way that gives names to the traversed edges or shows where certain known edges are located on the walk, like S “ pv0, e1, v1, . . . , ek, vkq where ei`1 “ {vi, vi`1} P E for all 0 ď i ă k and so EpSq “ {e1, . . . , ek}. For v P V , defineNpvq :“ {w P V ; {v, w} P E} the set of neighbors a.k.a. neighborhood of v. The degree of v is degpvq :“ |Npvq|. Given a set of verticesX Ď V , its neighborhood isNpXq :“ p⋃xPX NpxqqzX.