Planar Maximum Matching: Towards a Parallel Algorithm

Perfect matchings in planar graphs have been extensively studied and understood in the context of parallel complexity [21, 36, 25, 6, 2]. However, corresponding results for maximum matchings have been elusive. We partly bridge this gap by proving: 1. An SPL upper bound for planar bipartite maximum matching search. 2. Planar maximum matching search reduces to planar maximum matching decision. 3. Planar maximum matching count reduces to planar bipartite maximum matching count and planar maximum matching decision. The first bound improves on the known [18] bound of LC=L and is adaptable to any special bipartite graph class with non-zero circulation such as bounded genus graphs, K3,3-free graphs and K5-free graphs. Our bounds and reductions non-trivially combine techniques like the GallaiEdmonds decomposition [23], deterministic isolation [6, 7, 3], and the recent breakthroughs in the parallel search for planar perfect matchings [2, 32]. 2012 ACM Subject Classification Theory of computation → Parallel algorithms

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