Abstract. 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 Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or direct commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specific permission and/or a fee. Permissions may be requested from Publications Dept., ACM, Inc., 2 Penn Plaza, Suite 701 New York, NY 10121-0701 USA, fax +1 (212) 869-0481, or permissions@acm.org.  A General Approach to Online Network Optimization Problems in advance are the connectivity (bandwidth) or cut demands between vertices in the network which arrive online. We develop a unified framework for designing online algorithms for problems involving connectivity and cuts. We first present a general O(log m)-competitive deterministic algorithm for generating a fractional solution that satisfies the online connectivity or cut demands, where m is the number of edges in the graph. This may be of independent interest for solving fractional online bandwidth allocation problems, and is applicable to both directed and undirected graphs. We then show how to obtain integral solutions via an online rounding of the fractional solution. This part of the framework is problem dependent, and applies various tools including results on approximate max-flow min-cut for multicommodity flow, the Hierarchically Separated Trees (HST) method and its extensions, certain rounding techniques for dependent variables, and Räcke's new hierarchical decomposition of graphs. Specifically, our results for the integral case include an O(log m log n)-competitive randomized algorithm for the online nonmetric facility location problem and for a generalization of the problem called the multicast problem. In the nonmetric facility location problem, m is the number of facilities and n is the number of clients. The competitive ratio is nearly tight. We also present an O(log 2 n log k)-competitive randomized algorithm for the online group Steiner problem in trees and an O(log 3 n log k)-competitive randomized algorithm for the problem in general graphs, where n is the number of vertices in the graph and k is the number of groups. Finally, we design a deterministic O(log 3 n log log n)-competitive algorithm for the online multi-cut problem.
In REF , the first phase is performed by a general O(log n)-competitive method which is applicable to a wide class of online graph optimization problems.
A general approach to online network optimization problems
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