On partitioning graphs via single commodity flows

Authors:
Lorenzo Orecchia;Leonard J. Schulman;Umesh V. Vazirani;Nisheeth K. Vishnoi
Affiliations:
University of California, Berkeley, Berkeley, USA;California Institute of Technology, Pasadena, USA;University of California, Berkeley, Berkeley, USA;IBM India Research Lab, New Delhi, India
Venue:
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Year:
2008

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Abstract

In this paper we obtain improved upper and lower bounds for the best approximation factor for Sparsest Cut achievable in the cut-matching game framework proposed in Khandekar et al. [9]. We show that this simple framework can be used to design combinatorial algorithms that achieve O(log n) approximation factor and whose running time is dominated by a poly-logarithmic number of single-commodity max-flow computations. This matches the performance of the algorithm of Arora and Kale [2]. Moreover, we also show that it is impossible to get an approximation factor of better than Ω(√log n) in the cut-matching game framework. These results suggest that the simple and concrete abstraction of the cut-matching game may be powerful enough to capture the essential features of the complexity of Sparsest Cut.