Routing under balance

We introduce the notion of balance for directed graphs: a weighted directed graph is α-balanced if for every cut S ⊆ V, the total weight of edges going from S to V∖ S is within factor α of the total weight of edges going from V∖ S to S. Several important families of graphs are nearly balanced, in particular, Eulerian graphs (with α = 1) and residual graphs of (1+є)-approximate undirected maximum flows (with α=O(1/є)). We use the notion of balance to give a more fine-grained understanding of several well-studied routing questions that are considerably harder in directed graphs. We first revisit oblivious routings in directed graphs. Our main algorithmic result is an oblivious routing scheme for single-source instances that achieve an O(α · log3 n / loglogn) competitive ratio. In the process, we make several technical contributions which may be of independent interest. In particular, we give an efficient algorithm for computing low-radius decompositions of directed graphs parameterized by balance. We also define and construct low-stretch arborescences, a generalization of low-stretch spanning trees to directed graphs. On the negative side, we present new lower bounds for oblivious routing problems on directed graphs. We show that the competitive ratio of oblivious routing algorithms for directed graphs is Ω(n) in general; this result improves upon the long-standing best known lower bound of Ω(√n) by Hajiaghayi et al. We also show that our restriction to single-source instances is necessary by showing an Ω(√n) lower bound for multiple-source oblivious routing in Eulerian graphs. We also study the maximum flow problem in balanced directed graphs with arbitrary capacities. We develop an efficient algorithm that finds an (1+є)-approximate maximum flows in α-balanced graphs in time O(m α2 / є2). We show that, using our approximate maximum flow algorithm, we can efficiently determine whether a given directed graph is α-balanced. Additionally, we give an application to the directed sparsest cut problem.

[1]  Aleksander Madry,et al.  Navigating Central Path with Electrical Flows: From Flows to Matchings, and Back , 2013, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.

[2]  Marcin Bienkowski,et al.  A practical algorithm for constructing oblivious routing schemes , 2003, SPAA '03.

[3]  Anand Louis Cut-Matching Games on Directed Graphs , 2010, ArXiv.

[4]  Gary L. Miller,et al.  Parallel graph decompositions using random shifts , 2013, SPAA.

[5]  Peter Richtárik,et al.  Iteration complexity of randomized block-coordinate descent methods for minimizing a composite function , 2011, Mathematical Programming.

[6]  Mohammad Taghi Hajiaghayi,et al.  Oblivious routing in directed graphs with random demands , 2005, STOC '05.

[7]  Stefano Leonardi,et al.  On-Line Routing in All-Optical Networks , 1997, ICALP.

[8]  Harald Räcke,et al.  Oblivious Routing for the Lp-norm , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[9]  Mohammad Taghi Hajiaghayi,et al.  New lower bounds for oblivious routing in undirected graphs , 2006, SODA '06.

[10]  Jakub W. Pachocki,et al.  Stretching Stretch , 2014, ArXiv.

[11]  Baruch Awerbuch,et al.  Complexity of network synchronization , 1985, JACM.

[12]  Harald Räcke,et al.  Optimal hierarchical decompositions for congestion minimization in networks , 2008, STOC.

[13]  Jonah Sherman,et al.  Nearly Maximum Flows in Nearly Linear Time , 2013, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.

[14]  Aleksander Madry,et al.  Fast Approximation Algorithms for Cut-Based Problems in Undirected Graphs , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[15]  Yin Tat Lee,et al.  Matching the Universal Barrier Without Paying the Costs : Solving Linear Programs with Õ(sqrt(rank)) Linear System Solves , 2013, ArXiv.

[16]  Yin Tat Lee,et al.  An Almost-Linear-Time Algorithm for Approximate Max Flow in Undirected Graphs, and its Multicommodity Generalizations , 2013, SODA.

[17]  Mohammad Taghi Hajiaghayi,et al.  Oblivious routing on node-capacitated and directed graphs , 2005, SODA '05.

[18]  Richard Peng A Note on Cut-Approximators and Approximating Undirected Max Flows , 2014, ArXiv.

[19]  Andrew V. Goldberg,et al.  Beyond the flow decomposition barrier , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[20]  Yair Bartal,et al.  Probabilistic approximation of metric spaces and its algorithmic applications , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[21]  Noga Alon,et al.  A Graph-Theoretic Game and Its Application to the k-Server Problem , 1995, SIAM J. Comput..

[22]  Bruce M. Maggs,et al.  Exploiting locality for data management in systems of limited bandwidth , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[23]  Harald Räcke,et al.  Minimizing Congestion in General Networks , 2002, FOCS.

[24]  Chintan Shah,et al.  Computing Cut-Based Hierarchical Decompositions in Almost Linear Time , 2014, SODA.

[25]  Edith Cohen,et al.  Optimal oblivious routing in polynomial time , 2004, J. Comput. Syst. Sci..

[26]  Yurii Nesterov,et al.  Gradient methods for minimizing composite functions , 2012, Mathematical Programming.

[27]  Harald Räcke Survey on Oblivious Routing Strategies , 2009, CiE.

[28]  Jakub W. Pachocki,et al.  Solving SDD linear systems in nearly mlog1/2n time , 2014, STOC.