Complexity of finding embeddings in a k-tree
SIAM Journal on Algebraic and Discrete Methods
Easy problems for tree-decomposable graphs
Journal of Algorithms
Network flows: theory, algorithms, and applications
Network flows: theory, algorithms, and applications
A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth
SIAM Journal on Computing
Characterizations of k-terminal flow networks and computing network flows in partial k-trees
Proceedings of the sixth annual ACM-SIAM symposium on Discrete algorithms
Algorithms for fault-tolerant routing in circuit switched networks
Proceedings of the fourteenth annual ACM symposium on Parallel algorithms and architectures
Treewidth: Algorithmoc Techniques and Results
MFCS '97 Proceedings of the 22nd International Symposium on Mathematical Foundations of Computer Science
Dynamic Programming on Graphs with Bounded Treewidth
ICALP '88 Proceedings of the 15th International Colloquium on Automata, Languages and Programming
Approximation algorithms for disjoint paths problems
Approximation algorithms for disjoint paths problems
Complexity and approximability of k-splittable flows
Theoretical Computer Science
On the approximation of the single source k-splittable flow problem
Journal of Discrete Algorithms
A Branch and Price algorithm for the k-splittable maximum flow problem
Discrete Optimization
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Given a graph with a source and a sink node, the NP–hard maximum k–splittable flow (MkSF) problem is to find a flow of maximum value with a flow decomposition using at most k paths [6]. The multicommodity variant of this problem is a natural generalization of disjoint paths and unsplittable flow problems. Constructing a k–splittable flow requires two interdepending decisions. One has to decide on k paths (routing) and on the flow values on these paths (packing). We give efficient algorithms for computing exact and approximate solutions by decoupling the two decisions into a first packing step and a second routing step. Our main contributions are as follows: – We show that for constant k a polynomial number of packing alternatives containing at least one packing used by an optimal MkSF solution can be constructed in polynomial time. If k is part of the input, we obtain a slightly weaker result. In this case we can guarantee that, for any fixed ε0, the computed set of alternatives contains a packing used by a (1–ε)–approximate solution. The latter result is based on the observation that (1–ε)–approximate flows only require constantly many different flow values. We believe that this observation is of interest in its own right. – Based on (i), we prove that, for constant k, the MkSF problem can be solved in polynomial time on graphs of bounded treewidth. If k is part of the input, this problem is still NP–hard and we present a polynomial time approximation scheme for it. – Finally, we provide a comprehensive overview of the complexity and approximability landscape of MkSF for different values of k.