A general approach to online network optimization problems

We study a wide range of online graph and network optimization problems, focusing on problems that arise in the study of connectivity and cuts in graphs. In a general online network design problem, we have a communication network known to the algorithm in advance. What is not known in advance are the bandwidth or cut demands between nodes in the network. Our results include an <i>O</i>(log <i>m</i> log <i>n</i>) competitive randomized algorithm for the online non-metric facility location and for a generalization of the problem called themulticast problem. In the non-metric facility location <i>m</i> is the number of facilities and <i>n</i> is the number of clients. The competitive ratio is nearly tight. We also present an<i>O</i>(log<sup>2</sup> <i>n</i> log <i>k</i>) competitive randomized algorithm for the on-line group Steiner problem in trees and an <i>O</i>(log<sup>3</sup> <i>n</i> log <i>k</i>)competitive randomized algorithm for the problem in general graphs, where <i>n</i> is the number of vertices in the graph and <i>k</i> is the number of groups. Finally, we design a deterministic <i>O</i>(log<sup>3</sup> <i>n</i> log log <i>n</i>) competitive algorithm for the online multi-cut problem. Our algorithms are based on a unified framework for designing online algorithms for problems involving connectivity and cuts. We first present a general <i>O</i>(log <i>m</i>)-deterministic algorithm for generating fractional solution that satisfies the online connectivity or cut demands, where <i>m</i> is the number of edges in the graph.

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

[2]  Noga Alon,et al.  The online set cover problem , 2003, STOC '03.

[3]  Satish Rao,et al.  A tight bound on approximating arbitrary metrics by tree metrics , 2003, STOC '03.

[4]  Vijay V. Vazirani,et al.  Approximation Algorithms , 2001, Springer Berlin Heidelberg.

[5]  Piotr Berman,et al.  On-line algorithms for Steiner tree problems (extended abstract) , 1997, STOC '97.

[6]  Mihalis Yannakakis,et al.  Primal-dual approximation algorithms for integral flow and multicut in trees , 1997, Algorithmica.

[7]  Éva Tardos,et al.  Fast Approximation Algorithms for Fractional Packing and Covering Problems , 1995, Math. Oper. Res..

[8]  N. Alon,et al.  The Probabilistic Method, Second Edition , 2000 .

[9]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[10]  Noga Alon,et al.  The Probabilistic Method, Second Edition , 2004 .

[11]  Mihalis Yannakakis,et al.  Approximate Max-Flow Min-(Multi)Cut Theorems and Their Applications , 1996, SIAM J. Comput..

[12]  Satish Rao,et al.  A polynomial-time tree decomposition to minimize congestion , 2003, SPAA '03.

[13]  Makoto Imase,et al.  Dynamic Steiner Tree Problem , 1991, SIAM J. Discret. Math..

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

[15]  Yossi Azar,et al.  On-line generalized Steiner problem , 1996, SODA '96.

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

[17]  R. Ravi,et al.  A polylogarithmic approximation algorithm for the group Steiner tree problem , 2000, SODA '98.

[18]  Adam Meyerson,et al.  Online facility location , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[19]  Dimitris Fotakis,et al.  On the Competitive Ratio for Online Facility Location , 2003, Algorithmica.

[20]  Dorit S. Hochbaum,et al.  Approximation Algorithms for NP-Hard Problems , 1996 .