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Distributed Maximum Flow in Planar Graphs

Abstract

The dual of a planar graph GG is a planar graph GG^* that has a vertex for each face of GG and an edge for each pair of adjacent faces of GG. The profound relationship between a planar graph and its dual has been the algorithmic basis for solving numerous (centralized) classical problems on planar graphs. In the distributed setting however, the only use of planar duality is for finding a recursive decomposition of GG [DISC 2017, STOC 2019]. We extend the distributed algorithmic toolkit to work on the dual graph GG^*. These tools can then facilitate various algorithms on GG by solving a suitable dual problem on GG^*. Given a directed planar graph GG with positive and negative edge-lengths and hop-diameter DD, our key result is an O~(D2)\tilde{O}(D^2)-round algorithm for Single Source Shortest Paths on GG^*, which then implies an O~(D2)\tilde{O}(D^2)-round algorithm for Maximum stst-Flow on GG. Prior to our work, no O~(poly(D))\tilde{O}(\text{poly}(D))-round algorithm was known for Maximum stst-Flow. We further obtain a Dno(1)D\cdot n^{o(1)}-rounds (1ϵ)(1-\epsilon)-approximation algorithm for Maximum stst-Flow on GG when GG is undirected and stst-planar. Finally, we give a near optimal O~(D)\tilde O(D)-round algorithm for computing the weighted girth of GG. The main challenges in our work are that GG^* is not the communication graph (e.g., a vertex of GG is mapped to multiple vertices of GG^*), and that the diameter of GG^* can be much larger than DD (i.e., possibly by a linear factor). We overcome these challenges by carefully defining and maintaining subgraphs of the dual graph GG^* while applying the recursive decomposition on the primal graph GG. The main technical difficulty, is that along the recursive decomposition, a face of GG gets shattered into (disconnected) components yet we still need to treat it as a dual node.

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