Fast Algorithms for Directed Graph Partitioning Using Flows and Reweighted Eigenvalues

• URL: http://arxiv.org/abs/2306.09128v1
• Date: Thu, 15 Jun 2023 13:41:17 GMT
• Title: Fast Algorithms for Directed Graph Partitioning Using Flows and Reweighted Eigenvalues
• Authors: Lap Chi Lau, Kam Chuen Tung, Robert Wang
• Abstract summary: We derive almost linear-time algorithms to achieve $O(sqrtlogn)$-approximation and Cheeger-type guarantee for directed edge expansion. This provides a primal-dual flow-based framework to obtain the best known algorithms for directed graph partitioning.
• Score: 6.094384342913063
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider a new semidefinite programming relaxation for directed edge expansion, which is obtained by adding triangle inequalities to the reweighted eigenvalue formulation. Applying the matrix multiplicative weight update method to this relaxation, we derive almost linear-time algorithms to achieve $O(\sqrt{\log{n}})$-approximation and Cheeger-type guarantee for directed edge expansion, as well as an improved cut-matching game for directed graphs. This provides a primal-dual flow-based framework to obtain the best known algorithms for directed graph partitioning. The same approach also works for vertex expansion and for hypergraphs, providing a simple and unified approach to achieve the best known results for different expansion problems and different algorithmic techniques.

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