A framework for dynamic matching in weighted graphs

We introduce a new framework for computing approximate maximum weight matchings. Our primary focus is on the fully dynamic setting, where there is a large gap between the guarantees of the best known algorithms for computing weighted and unweighted matchings. Indeed, almost all current weighted matching algorithms that reduce to the unweighted problem lose a factor of two in the approximation ratio. In contrast, in other sublinear models such as the distributed and streaming models, recent work has largely closed this weighted/unweighted gap. For bipartite graphs, we almost completely settle the gap with a general reduction that converts any algorithm for α-approximate unweighted matching to an algorithm for (1−)α-approximate weighted matching, while only increasing the update time by an O (log n ) factor for constant . We also show that our framework leads to significant improvements for non-bipartite graphs, though not in the form of a universal reduction. In particular, we give two algorithms for weighted non-bipartite matching: 1. A randomized (Las Vegas) fully dynamic algorithm that maintains a (1/2−)-approximate maximum weight matching in worst-case update time O (polylog n ) with high probability against an adaptive adversary. Our bounds are essentially the same as those of the unweighted algorithm of Wajc [STOC 2020]. 2. A deterministic fully dynamic algorithm that maintains a (2/3−)-approximate maximum weight matching in amortized update time O ( m 1/4 ). Our bounds are essentially the same as those of the unweighted algorithm of Bernstein and Stein [SODA 2016]. A key feature of our framework is that it uses existing algorithms for unweighted matching as black-boxes. As a result, our framework is simple and versatile. Moreover, our framework easily translates to other models, and we use it to derive new results for the weighted matching problem in streaming and communication complexity models.

Authors

• Aaron Bernstein
• Aditi Dudeja
• Zachary Langley

Keywords

• Adaptive Adversary
• Dynamic Algorithms
• Dynamic Matching
• Matching Sparsifiers
• Weighted Matching