Blossom VI: A Practical Minimum Weight Perfect Matching Algorithm

Minimum weight perfect matching is a fundamental problem in combinatorial optimization. Since 2009, Blossom V has been the leading practical implementation. We present Blossom VI, a practical algorithm that improves upon Blossom V. We test the performance of our implementation on nine benchmark families with graphs containing up to roughly 10 million edges. Blossom VI is significantly faster on the hardest tested instances, while being at most a factor of two slower on families where Blossom V already scales nearly linearly. Blossom VI is a primal-dual solver whose primal phase computes a maximum-cardinality matching in the zero-slack subgraph using cherry trees. At the end of each primal phase, Blossom VI contracts entire cherry blossoms instead of sequences of nested traditional blossoms that would be created by Blossom V. This produces shallower contraction hierarchies and avoids costly cascades of blossom expansions. Our measurements show that reduced blossom depth and fewer expensive expansion operations can explain the observed improvement.