patmorin boosted

New post: Orientations of infinite graphs, 11011110.github.io/blog/2019/0

In many cases the existence of an orientation with desired properties can be reduced to its existence in finite subgraphs, using a mathod from Rado (1949).

Recently, Bekos et al made a major breakthrough, showing that planar graphs of bounded degree have bounded queue number.

arxiv.org/abs/1811.00816

We were able to generalize this to bounded degree bounded genus graphs.

arxiv.org/pdf/1901.05594.pdf

A recent arXiv preprint on robust spanners. For any \(n\)-point set in \(\mathbb{R}ᵈ\), we show how to construct spanners with \(O(n\log²n\log\log n)\) edges so that, if you remove \(k\) vertices, the remaining graph is a spanner of \(n-(1+ε)k\) of its vertices.

arxiv.org/abs/1812.09913

We called these \((1+ε)k\)-robust spanners. Buchin, Har-Peled, and Olah have a much catchier name: 'a spanner for the day after'

arxiv.org/abs/1811.06898

patmorin boosted

New post: Gurobi versus the no-three-in-line problem
11011110.github.io/blog/2018/1

How well can a good but generic integer linear program solver stand up against 20-year-old problem-specific search code?

Two new arXiv preprints:

arxiv.org/abs/1811.03432
arxiv.org/abs/1811.03427

The first shows that if a graph \(G\) has a non-crossing straight-line drawing (a.k.a. Fáry drawing) in which some set \(S\) of vertices is drawn on a single line then, for any point set \(X\) of size \(|S|\), \(G\) has a Fáry drawing in which the vertices in \(S\) are drawn on the points in \(X\).

The second shows that, in bounded degree planar graphs, one can always find such a set \(S\) of size \(\Omega(n^{0.8})\).

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