Generalizations of the Kolmogorov-Barzdin embedding estimates

We consider several ways to measure the `geometric complexity' of an embedding from a simplicial complex into Euclidean space. One of these is a version of `thickness', based on a paper of Kolmogorov and Barzdin. We prove inequalities relating the thickness and the number of simplices in the simplicial complex, generalizing an estimate that Kolmogorov and Barzdin proved for graphs. We also consider the distortion of knots. We give an alternate proof of a theorem of Pardon that there are isotopy classes of knots requiring arbitrarily large distortion. This proof is based on the expander-like properties of arithmetic hyperbolic manifolds.Comment: 45 page

Hyperbolic intersection graphs and (quasi)-polynomial time

We study unit ball graphs (and, more generally, so-called noisy uniform ball graphs) in $d$-dimensional hyperbolic space, which we denote by $\mathbb{H}^d$. Using a new separator theorem, we show that unit ball graphs in $\mathbb{H}^d$ enjoy similar properties as their Euclidean counterparts, but in one dimension lower: many standard graph problems, such as Independent Set, Dominating Set, Steiner Tree, and Hamiltonian Cycle can be solved in $2^{O(n^{1-1/(d-1)})}$ time for any fixed $d\geq 3$, while the same problems need $2^{O(n^{1-1/d})}$ time in $\mathbb{R}^d$. We also show that these algorithms in $\mathbb{H}^d$ are optimal up to constant factors in the exponent under ETH. This drop in dimension has the largest impact in $\mathbb{H}^2$, where we introduce a new technique to bound the treewidth of noisy uniform disk graphs. The bounds yield quasi-polynomial ($n^{O(\log n)}$) algorithms for all of the studied problems, while in the case of Hamiltonian Cycle and $3$-Coloring we even get polynomial time algorithms. Furthermore, if the underlying noisy disks in $\mathbb{H}^2$ have constant maximum degree, then all studied problems can be solved in polynomial time. This contrasts with the fact that these problems require $2^{\Omega(\sqrt{n})}$ time under ETH in constant maximum degree Euclidean unit disk graphs. Finally, we complement our quasi-polynomial algorithm for Independent Set in noisy uniform disk graphs with a matching $n^{\Omega(\log n)}$ lower bound under ETH. This shows that the hyperbolic plane is a potential source of NP-intermediate problems.Comment: Short version appears in SODA 202

Matroids with nine elements

We describe the computation of a catalogue containing all matroids with up to nine elements, and present some fundamental data arising from this cataogue. Our computation confirms and extends the results obtained in the 1960s by Blackburn, Crapo and Higgs. The matroids and associated data are stored in an online database, and we give three short examples of the use of this database.Comment: 22 page