Network flows: theory, algorithms, and applications
Network flows: theory, algorithms, and applications
Algorithms for fault-tolerant routing in circuit switched networks
Proceedings of the fourteenth annual ACM symposium on Parallel algorithms and architectures
Approximation algorithms for disjoint paths problems
Approximation algorithms for disjoint paths problems
Approximation and complexity of k–splittable flows
WAOA'05 Proceedings of the Third international conference on Approximation and Online Algorithms
Maximum flows on disjoint paths
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
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Let G = (V, E) be a graph with a source node s and a sink node t, |V| = n, |E| = m. For a given number k, the Maximum k-Splittable Flow problem (MkSF) is to find an s, t-flow of maximum value with a flow decomposition using at most k paths. In the multicommodity case this problem generalizes disjoint paths problems and unsplittable flow problems. We provide a comprehensive overview of the complexity and approximability landscape of MkSF on directed and undirected graphs. We consider constant values of k and k depending on graph parameters. For arbitrary constant values of k, we prove that the problem is strongly NP-hard on directed and undirected graphs already for k = 2. This extends a known NP-hardness result for directed graphs that could not be applied to undirected graphs. Furthermore, we show that MkSF cannot be approximated with a performance ratio better than 5/6. This is the first constant bound given for arbitrary constant values of k. For non-constant values of k, the polynomial solvability was known before for all k ≥ m, but open for smaller k. We prove that MkSF is NP-hard for all k fulfilling 2 ≤ k ≤ m - n + 1 (for n ≥ 3). For all other values of k the problem is shown to be polynomially solvable.