Distributed Maximum Flow in Planar Graphs

2024-11-18

arXiv.org

AI Summary

This paper addresses three classical problems on planar graphs in the distributed CONGEST model: maximum $s$-$t$ flow, directed global minimum cut, and weighted girth. It introduces the dual graph $G^*$ systematically into distributed algorithm design for the first time. Methodologically, it constructs and dynamically maintains $G^*$ under non-communication graph constraints, and integrates heavy-light decomposition with localized synchronization among dual nodes to resolve connectivity and diameter mismatches caused by face splitting. Theoretical contributions include: (i) $ ilde{O}(D^2)$-round exact algorithms for maximum $s$-$t$ flow and directed global minimum cut; (ii) a $D cdot n^{o(1)}$-round $(1-varepsilon)$-approximation for maximum $s$-$t$ flow on undirected $s$-$t$-planar graphs; and (iii) a near-optimal $ ilde{O}(D)$-round algorithm for weighted girth computation. This work establishes the first distributed optimization framework for planar graphs grounded in duality.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The dual of a planar graph $G$ is a planar graph $G^*$ that has a vertex for each face of $G$ and an edge for each pair of adjacent faces of $G$. 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 $G$ [DISC 2017, STOC 2019]. We extend the distributed algorithmic toolkit to work on the dual graph $G^*$. These tools can then facilitate various algorithms on $G$ by solving a suitable dual problem on $G^*$. Given a directed planar graph $G$ with positive and negative edge-lengths and hop-diameter $D$, our key result is an $ ilde{O}(D^2)$-round algorithm for Single Source Shortest Paths on $G^*$, which then implies $ ilde{O}(D^2)$-round algorithms for Maximum $st$-Flow and Directed Global Min-Cut on $G$. Prior to our work, no $ ilde{O}( ext{poly}(D))$-round algorithm was known for those problems. We further obtain a $Dcdot n^{o(1)}$-rounds $(1-epsilon)$-approximation algorithm for Maximum $st$-Flow on $G$ when $G$ is undirected and $st$-planar. Finally, we give a near optimal $ ilde O(D)$-round algorithm for computing the weighted girth of $G$. The main challenges in our work are that $G^*$ is not the communication graph (e.g., a vertex of $G$ is mapped to multiple vertices of $G^*$), and that the diameter of $G^*$ can be much larger than $D$ (i.e., possibly by a linear factor). We overcome these challenges by carefully defining and maintaining subgraphs of the dual graph $G^*$ while applying the recursive decomposition on the primal graph $G$. The main technical difficulty, is that along the recursive decomposition, a face of $G$ gets shattered into (disconnected) components yet we still need to treat it as a dual node.

Problem

• Extends distributed tools to solve problems on dual planar graphs
• Develops efficient algorithms for Maximum Flow and Min-Cut
• Addresses challenges in dual graph diameter and communication

Innovation

• Extends distributed toolkit to dual graph G*
• Provides tilde{O}(D^2)-round SSSP on G*
• Overcomes challenges via recursive decomposition