Improved deterministic distributed matching via rounding

We present improved deterministic distributed algorithms for a number of well-studied matching problems, which are simpler, faster, more accurate, and/or more general than their known counterparts. The common denominator of these results is a deterministic distributed rounding method for certain linear programs, which is the first such rounding method, to our knowledge. A sampling of our end results is as follows: An \(O\mathopen {}\left( \log ^2 \Delta \cdot \log n\right) \mathclose {}\)-round deterministic distributed algorithm for computing a maximal matching, in n-node graphs with maximum degree \(\Delta \). This is the first improvement in about 20 years over the celebrated \(O(\log ^4 n)\)-round algorithm of Hanckowiak, Karonski, and Panconesi [SODA’98, PODC’99]. A deterministic distributed algorithm for computing a \((2+\varepsilon )\)-approximation of maximum matching in \(O\mathopen {}\left( \log ^2 \Delta \cdot \log \frac{1}{\varepsilon } + \log ^ * n\right) \mathclose {}\) rounds. This is exponentially faster than the classic \(O(\Delta +\log ^* n)\)-round 2-approximation of Panconesi and Rizzi [DIST’01]. With some modifications, the algorithm can also find an almost maximal matching which leaves only an \(\varepsilon \)-fraction of the edges on unmatched nodes. An \(O\mathopen {}\left( \log ^2 \Delta \cdot \log \frac{1}{\varepsilon } \cdot \log _{1+\varepsilon } W + \log ^ * n\right) \mathclose {}\)-round deterministic distributed algorithm for computing a \((2+\varepsilon )\)-approximation of a maximum weighted matching, and also for the more general problem of maximum weighted b-matching. Here, W denotes the maximum normalized weight. These improve over the \(O\mathopen {}\left( \log ^4 n \cdot \log _{1+\varepsilon } W\right) \mathclose {}\)-round \((6+\varepsilon )\)-approximation algorithm of Panconesi and Sozio [DIST’10].