Coarse Differentiation and Multi-flows in Planar Graphs

@article{Lee2008CoarseDA,
  title={Coarse Differentiation and Multi-flows in Planar Graphs},
  author={James R. Lee and Prasad Raghavendra},
  journal={Discrete \& Computational Geometry},
  year={2008},
  volume={43},
  pages={346-362}
}
AbstractWe show that the multi-commodity max-flow/min-cut gap for series-parallel graphs can be as bad as 2, matching a recent upper bound (Chakrabarti et al. in 49th Annual Symposium on Foundations of Computer Science, pp. 761–770, 2008) for this class, and resolving one side of a conjecture of Gupta, Newman, Rabinovich, and Sinclair.This also improves the largest known gap for planar graphs from $\frac{3}{2}$ to 2, yielding the first lower bound that does not follow from elementary… Expand
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References

SHOWING 1-10 OF 40 REFERENCES
Expander flows, geometric embeddings and graph partitioning
TLDR
An interesting and natural "certificate" for a graph's expansion is described, by embedding an n-node expander in it with appropriate dilation and congestion, and is called an expander flow. Expand
On multicommodity flows in planar graphs
TLDR
It is shown, by presenting counterexamples, that the half-integrality property does not necessarily hold when either the graph cannot be drawn in the plane with all sources and sinks on a common face, or the graph is directed. Expand
The geometry of graphs and some of its algorithmic applications
TLDR
Efficient algorithms for embedding graphs low-dimensionally with a small distortion, and a new deterministic polynomial-time algorithm that finds a (nearly tight) cut meeting this bound. Expand
Vertex cuts, random walks, and dimension reduction in series-parallel graphs
TLDR
It is shown that series-parallel metrics have Markov type 2, which generalizes a result of Naor, Peres, Schramm, and Sheffield for trees and yields an O(√log n) approximation algorithm for vertex sparsestcut in such graphs, as well as an O(*log k) approximate max-flow/min-vertex-cut theorem for series- parallel instances with<i>k</i> terminals. Expand
Embedding k-outerplanar graphs into ℓ1
TLDR
It is shown that the shortest-path metric of any k-outerplanar graph, for any fixed k, can be approximated by a probability distribution over tree metrics with constant distortion, and hence also embedded into l1 with constant distort, implying a constant upper bound on the ratio between the sparsest cut and the maximum concurrent flow in multicommodity networks for k- outerplanar graphs. Expand
Cuts, Trees and ℓ1-Embeddings of Graphs*
TLDR
It is shown, surprisingly, that such metrics approximate distances very poorly even for families of graphs with low treewidth, and excludes the possibility of using them to explore the finer structure of ℓ1-embeddability. Expand
The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative Type Metrics into l1
TLDR
This paper disproves the non-uniform version of Arora, Rao and Vazirani's Conjecture (2004), asserting that the integrality gap of the sparsest cut SDP, with the triangle inequality constraints, is bounded from above by a constant. Expand
Geometry of cuts and metrics
TLDR
This book draws from the interdisciplinarity of these fields as it gathers methods and results from polytope theory, geometry of numbers, probability theory, design and graph theory around two objects, cuts and metrics. Expand
Embeddings of Topological Graphs: Lossy Invariants, Linearization, and 2-Sums
We study the properties of embeddings, multicommodity flows, and sparse cuts in minor-closed families of graphs which are also closed under 2-sums; this includes planar graphs, graphs of boundedExpand
An O(log k) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm
TLDR
It is shown that the minimum cut ratio is within a factor of O(log k) of the maximum concurrent flow for k-commodity flow instances with arbitrary capacities and demands, and thus of the optimal min-cut ratio. Expand
...
1
2
3
4
...