Abstract
In this paper, we mainly consider the complexity of the k-splittable flow minimizing congestion problem. We give some complexity results. For the k-splittable flow problem, the existence of a feasible solution is strongly NP-hard. When the number of the source nodes is an input, for the uniformly exactly k-splittable flow problem, obtaining an approximation algorithm with performance ratio better than (√5+1)/2 is NP-hard. When k is an input, for single commodity k-splittable flow problem, obtaining an algorithm with performance ratio better than is NP-hard. In the last of the paper, we study the relationship of minimizing congestion and minimizing number of rounds in the k-splittable flow problem. The smaller the congestion is, the smaller the number of rounds.
Highlights
In the traditional multi-commodity flow problems, flow being sent from the source nodes to the destination nodes may travel on large number of paths through the network
We mainly consider the complexity of the k-splittable flow minimizing congestion problem
In the last of the paper, we study the relationship of minimizing congestion and minimizing number of rounds in the k-splittable flow problem
Summary
In the traditional multi-commodity flow problems, flow being sent from the source nodes to the destination nodes may travel on large number of paths through the network. For the unsplittable flow problem, Erlebach et al [3] proved that for arbitrary ε , obtaining an approximation algorithm for minimizing congestion and cost with performance ratio better than (2 − ε ,1) is NP-hard. Baier et al [7] solved the maximum Single- and Multi-commodity k-splittable flow problem using approximation algorithms. Kolliopoulos [9] studied the approximation algorithms for the single source 2-splittable flow problem using rounding down strategy and Salazar et al [10] considered the single. When the number of the source nodes is an input, for the uniformly exactly k-splittable flow problem, obtaining an approximation algorithm with performance ratio better than 2 is NP-hard. We study the relationship between minimum congestion and minimum number of rounds
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