A node-capacitated Okamura–Seymour theorem

@article{Lee2015ANO,
  title={A node-capacitated Okamura–Seymour theorem},
  author={James R. Lee and M. Mendel and Mohammadreza Moharrami},
  journal={Mathematical Programming},
  year={2015},
  volume={153},
  pages={381-415}
}
The classical Okamura–Seymour theorem states that for an edge-capacitated, multi-commodity flow instance in which all terminals lie on a single face of a planar graph, there exists a feasible concurrent flow if and only if the cut conditions are satisfied. Simple examples show that a similar theorem is impossible in the node-capacitated setting. Nevertheless, we prove that an approximate flow/cut theorem does hold: For some universal $$\varepsilon > 0$$ε>0, if the node cut conditions are… Expand
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References

SHOWING 1-10 OF 26 REFERENCES
Node-Capacitated Ring Routing
TLDR
It is proved that, independent of the integrality of node capacities, it suffices to require the distance inequality only for distances arising from (0-1-2)-valued node weights, a requirement that will be called the double-cut condition. Expand
Multicommodity flows and cuts in polymatroidal networks
TLDR
This work considers multicommodity flow and cut problems in polymatroidal networks where there are submodular capacity constraints on the edges incident to a node and establishes poly-logarithmic flow-cut gap results in several scenarios that have been previously considered in the standard network flow models. Expand
Excluded minors, network decomposition, and multicommodity flow
In this paper we show that, given a graph and parameters 6 and r, we can find either a K,,. minor or an edge-cut of size O(mT/6) whose removal yields components of weak diameter O(T-26); i.e., everyExpand
Vertex Sparsifiers: New Results from Old Techniques
TLDR
Efficient algorithms for constructing a flow-sparsifier H that maintains congestion up to a factor ofO(log k/log log k) where k = |K| and a convex combination of trees over the terminals K that maintains congested graphs is given. Expand
Edge-Disjoint Paths in Planar Graphs with Constant Congestion
TLDR
An approximation with congestion 4 is obtained and an alternative decomposition that is specific to planar graphs is developed that produces instances that are Okamura-Seymour instances, which have the property that all terminals lie on a single face. Expand
Computing Maximal "Polymatroidal" Network Flows
TLDR
The number of augmentations required to achieve a maximal value flow is bounded by the cube of the number of arcs in the network, provided each successive augmentation is made along a shortest augmenting path, with ties between shortest paths broken by lexicography. Expand
Maximal Flow Through a Network
Introduction. The problem discussed in this paper was formulated by T. Harris as follows: "Consider a rail network connecting two cities by way of a number of intermediate cities, where each link ofExpand
Improved Approximation Algorithms for Minimum Weight Vertex Separators
TLDR
It is shown that embeddings into $L_1$ are insufficient but that the additional structure provided by many embedding theorems does suffice for the authors' purposes, and an optimal $O(\log k)$-approximate max-flow/min-vertex-cut theorem for arbitrary vertex-capacitated multicommodity flow instances on $k$ terminals is proved. Expand
Approximation algorithms for the 0-extension problem
TLDR
It is proved that the integrality ratio of the metric relaxation is at least c√lgk for a positive c for infinitely many k and the results improve some of the results of Kleinberg and Tardos and they further the understanding on how to use metric relaxations. Expand
Multicommodity flows in planar graphs
TLDR
This paper solves the problem of when is there a flow for each i, between s i and t i and of value q i, such that the total flow through each edge does not exceed its capacity. Expand
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