An O(sqrt(|v|) |E|) Algorithm for Finding Maximum Matching in General Graphs

In this paper we present an 0(√|V|·|E|) algorithm for finding a maximum matching in general graphs. This algorithm works in 'phases'. In each phase a maximal set of disjoint minimum length augmenting paths is found, and the existing matching is increased along these paths. Our contribution consists in devising a special way of handling blossoms, which enables an O(|E|) implementation of a phase. In each phase, the algorithm grows Breadth First Search trees at all unmatched vertices. When it detects the presence of a blossom, it does not 'shrink' the blossom immediately. Instead, it delays the shrinking in such a way that the first augmenting path found is of minimum length. Furthermore, it achieves the effect of shrinking a blossom by a special labeling procedure which enables it to find an augmenting path through a blossom quickly.

Authors

• Silvio Micali
• Vijay V. Vazirani

Keywords

• Delay
• Labeling
• History
• Scholarships
• Flowering
• First Search
• Tree Search
• Breadth-first Search
• Maximum Matching
• Graph Matching
• Active Center
• Root Of The Tree
• Start Of Phase
• Incident Edges
• End Of Level