On Light Spanners, Low-treewidth Embeddings and Efficient Traversing in Minor-free Graphs

@article{CohenAddad2020OnLS,
title={On Light Spanners, Low-treewidth Embeddings and Efficient Traversing in Minor-free Graphs},
author={Vincent Cohen-Addad and Arnold Filtser and Philip N. Klein and Hung Le},
journal={2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)},
year={2020},
pages={589-600}
}
• Published 10 September 2020
• Computer Science, Mathematics
• 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
Understanding the structure of minor-free metrics, namely shortest path metrics obtained over a weighted graph excluding a fixed minor, has been an important research direction since the fundamental work of Robertson and Seymour. A fundamental idea that helps both to understand the structural properties of these metrics and lead to strong algorithmic results is to construct a “small-complexity” graph that approximately preserves distances between pairs of points of the metric. We show the two…

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References

SHOWING 1-10 OF 77 REFERENCES

• Mathematics
STOC
• 2014
We prove that any graph excluding Kr as a minor has can be partitioned into clusters of diameter at most Δ while removing at most O(r/Δ) fraction of the edges. This improves over the results of
• Computer Science, Mathematics
SPAA '07
• 2007
The first sparse covers and probabilistic partitions for graphs excluding a fixed minor that have strong diameter bounds are provided; i.e. each set of the cover/partition has a small diameter as an induced sub-graph.
• Mathematics, Computer Science
STOC
• 2018
The main result is an O(kmin{1p,12})-distortion embedding, which is a super-exponential improvement over the best previous bound of Lee and Sidiropoulos and implies improved distortion on bounded treewidth graphs (O((klogn)1p)).
We give the first PTAS for the subset Traveling Salesperson Problem (TSP) in $H$-minor-free graphs. This resolves a long standing open problem in a long line of work on designing PTASes for TSP in
• Mathematics, Computer Science
STOC '11
• 2011
We prove that any graph excluding a fixed minor can have its edges partitioned into a desired number k of color classes such that contracting the edges in any one color class results in a graph of
A decomposition theorem for graphs with excluded minors says that such graphs can be decomposed into trees of graphs of almost bounded local tree-width, and it is shown that a number of combinatorial optimization problems have a polynomial time approximation scheme when restricted to a class of graphs with an excluded minor.
• Mathematics
RANDOM-APPROX
• 2003
For any graph that excludes a K r,r minor, Klein, Plotkin and Rao showed that this can be done while cutting only O(r 3/δ) fraction of the edges, which implies a bound on multicommodity max-flow min-cut ratio for such graphs.
• Mathematics, Computer Science
ICALP
• 2011
For planar graphs, bounded-genus graphs, and minor-excluded graphs, this paper gives distance-oracle constructions that require only O(n) space, and the big O hides only a fixed constant, independent of e and independent of genus or size of an excluded minor.
• Mathematics
SIAM J. Discret. Math.
• 2012
The main result proves that f*(k)≥Ω(k2), significantly improving over the trivial f*[k]≥k, and the lower bound holds even for planar graphs G, in contrast to graphs G of constant treewidth, for which it is proved that O( k) vertices suffice.
• Mathematics
2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
• 2017
We show that every H-minor-free graph has a light (1+&#x2265;ilon)-spanner, resolving an open problem of Grigni and Sissokho and proving a conjecture of Grigni and Hung \cite{GH12}. Our lightness