Complexity of the Steiner Network Problem with Respect to the Number of Terminals

In the Directed Steiner Network problem we are given an arc-weighted digraph $G$, a set of terminals $T \subseteq V(G)$, and an (unweighted) directed request graph $R$ with $V(R)=T$. Our task is to output a subgraph $G' \subseteq G$ of the minimum cost such that there is a directed path from $s$ to $t$ in $G'$ for all $st \in A(R)$. It is known that the problem can be solved in time $|V(G)|^{O(|A(R)|)}$ [Feldman&Ruhl, SIAM J. Comput. 2006] and cannot be solved in time $|V(G)|^{o(|A(R)|)}$ even if $G$ is planar, unless Exponential-Time Hypothesis (ETH) fails [Chitnis et al., SODA 2014]. However, as this reduction (and other reductions showing hardness of the problem) only shows that the problem cannot be solved in time $|V(G)|^{o(|T|)}$ unless ETH fails, there is a significant gap in the complexity with respect to $|T|$ in the exponent. We show that Directed Steiner Network is solvable in time $f(R)\cdot |V(G)|^{O(c_g \cdot |T|)}$, where $c_g$ is a constant depending solely on the genus of $G$ and $f$ is a computable function. We complement this result by showing that there is no $f(R)\cdot |V(G)|^{o(|T|^2/ \log |T|)}$ algorithm for any function $f$ for the problem on general graphs, unless ETH fails

Kakeya-type sets in finite vector spaces

For a finite vector space $V$ and a non-negative integer $r\le\dim V$ we estimate the smallest possible size of a subset of $V$, containing a translate of every $r$-dimensional subspace. In particular, we show that if $K\subset V$ is the smallest subset with this property, $n$ denotes the dimension of $V$, and $q$ is the size of the underlying field, then for $r$ bounded and $r<n\le rq^{r-1}$ we have $|V\setminus K|=\Theta(nq^{n-r+1})$. This improves previously known bounds $|V\setminus K|=\Omega(q^{n-r+1})$ and $|V\setminus K|=O(n^2q^{n-r+1})$

A bound for the diameter of random hyperbolic graphs

Random hyperbolic graphs were recently introduced by Krioukov et. al. [KPKVB10] as a model for large networks. Gugelmann, Panagiotou, and Peter [GPP12] then initiated the rigorous study of random hyperbolic graphs using the following model: for $\alpha> \tfrac{1}{2}$, $C\in\mathbb{R}$, $n\in\mathbb{N}$, set $R=2\ln n+C$ and build the graph $G=(V,E)$ with $|V|=n$ as follows: For each $v\in V$, generate i.i.d. polar coordinates $(r_{v},\theta_{v})$ using the joint density function $f(r,\theta)$, with $\theta_{v}$ chosen uniformly from $[0,2\pi)$ and $r_{v}$ with density $f(r)=\frac{\alpha\sinh(\alpha r)}{\cosh(\alpha R)-1}$ for $0\leq r< R$. Then, join two vertices by an edge, if their hyperbolic distance is at most $R$. We prove that in the range $\tfrac{1}{2} < \alpha < 1$ a.a.s. for any two vertices of the same component, their graph distance is $O(\log^{C_0+1+o(1)}n)$, where $C_0=2/(\tfrac{1}{2}-\frac{3}{4}\alpha+\tfrac{\alpha^2}{4})$, thus answering a question raised in [GPP12] concerning the diameter of such random graphs. As a corollary from our proof we obtain that the second largest component has size $O(\log^{2C_0+1+o(1)}n)$, thus answering a question of Bode, Fountoulakis and M\"{u}ller [BFM13]. We also show that a.a.s. there exist isolated components forming a path of length $\Omega(\log n)$, thus yielding a lower bound on the size of the second largest component.Comment: 5 figure