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Flows on Few Paths: Algorithms and Lower Bounds
, 2004
"... Classical network flow theory allows decomposition of flow into several chunks of arbitrary sizes traveling through the network on different paths. In the first part of this paper we consider the unsplittable flow problem where all flow traveling from a source to a destination must be sent on only ..."
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Classical network flow theory allows decomposition of flow into several chunks of arbitrary sizes traveling through the network on different paths. In the first part of this paper we consider the unsplittable flow problem where all flow traveling from a source to a destination must be sent on only one path. We prove a lower bound of Ω(log m / log log m) on the performance of a general class of algorithms for minimizing congestion where m is the number of edges in a graph. These algorithms start with a solution for the classical multicommodity flow problem, compute a path decomposition, and select one of its paths for each commodity in order to obtain an unsplittable flow. Our result matches the best known upper bound for randomized rounding – an algorithm of this type introduced by Raghavan and Thompson. The ksplittable flow problem is a generalization of the unsplittable flow problem where the number of paths is bounded for each commodity. We study a new variant of this problem with additional constraints on the amount of flow being sent along each path. We present approximation results for two versions of this problem with the objective to minimize the congestion of the network. The key idea is to reduce the problem under consideration to an unsplittable flow problem while only losing a constant factor in the performance ratio.
Lengthbounded and dynamic ksplittable flows
 In Operations Research Proceedings 2005
, 2006
"... Summary. Classical network flow problems do not impose restrictions on the choice of paths on which flow is sent. Only the arc capacities of the network have to be obeyed. This scenario is not always realistic. In fact, there are many problems for which, e.g., the number of paths being used to route ..."
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Summary. Classical network flow problems do not impose restrictions on the choice of paths on which flow is sent. Only the arc capacities of the network have to be obeyed. This scenario is not always realistic. In fact, there are many problems for which, e.g., the number of paths being used to route a commodity or the length of such paths has to be small. These restrictions are considered in the lengthbounded ksplittable stflow problem: The problem is a variant of the well known classical stflow problem with the additional requirement that the number of paths that may be used to route the flow and the maximum length of those paths are bounded. Our main result is that we can efficiently compute a lengthbounded stflow which sends one fourth of the maximum flow value while exceeding the length bound by a factor of at most 2. We also show that this result leads to approximation algorithms for dynamic ksplittable stflows.