Directed metrics and directed graph partitioning problems

The theory of embeddings of finite metrics has provided a powerful toolkit for graph partitioning problems in undirected graphs. The connection comes from the fact that the integrality gaps of mathematical programming relaxations for sparsest cut in undirected graphs is exactly equal to the minimum distortion required to embed certain metrics into l1. No analog of this metric embedding theory exists for directed (asymmetric) metrics, the natural distance functions that arise in considering mathematical relaxations for directed graph partioning problems. We initiate a study of metric embeddings for directed metrics, motivated by understanding directed variants of sparsest cut.It turns out that there are two different ways to formulate sparsest cut in directed graphs (depending on whether one insists on partitioning the graph into two pieces or not). Different subclasses of directed metrics arise in the consideration of mathematical relaxations for these two formulations and the embedding questions that result are quite different. Unlike in the undirected case, where the natural host space is l1, the host space in the directed case is not obvious and depends on the problem formulation. Our work is a first step at understanding this space of directed metrics, the resulting embedding questions and their relationships to directed graph partitioning problems.

[1]  Yuval Rabani,et al.  Approximating Directed Multicuts , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[2]  James R. Lee,et al.  Euclidean distortion and the sparsest cut , 2005, STOC '05.

[3]  Shimon Kogan,et al.  Hardness of approximation of the Balanced Complete Bipartite Subgraph problem , 2004 .

[4]  J. Bourgain.  On lipschitz embedding of finite metric spaces in Hilbert space , 1985 .

[5]  Frank Thomson Leighton,et al.  An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[6]  Satish Rao,et al.  Expander flows, geometric embeddings and graph partitioning , 2004, STOC '04.

[7]  Amit Agarwal,et al.  O(√log n) approximation algorithms for min UnCut, min 2CNF deletion, and directed cut problems , 2005, STOC '05.

[8]  Uriel Feige,et al.  Resolution lower bounds for the weak pigeon hole principle , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[9]  Gert Vegter,et al.  In handbook of discrete and computational geometry , 1997 .

[10]  Anupam Gupta,et al.  Embeddings of negative-type metrics and an improved approximation to generalized sparsest cut , 2005, SODA '05.

[11]  M. Hajiaghayi,et al.  An O ( √ n )-Approximation Algorithm For Directed Sparsest Cut , 2005 .

[12]  Mohammad Taghi Hajiaghayi,et al.  An O(sqrt(n))-approximation algorithm for directed sparsest cut , 2006, Inf. Process. Lett..

[13]  Paolo Vitolo.  The representation of weighted quasimetric spaces , 1999 .

[14]  Mohammad Taghi Hajiaghayi,et al.  On the max-flow min-cut ratio for directed multicommodity flows , 2006, Theor. Comput. Sci..

[15]  Anupam Gupta.  Improved results for directed multicut , 2003, SODA '03.

[16]  Nisheeth K. Vishnoi,et al.  The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into l/sub 1/ , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).