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

## TL;DR

This paper investigates the computational complexity of the Directed Steiner Network problem, providing new algorithms for graphs with bounded genus and establishing tight hardness bounds related to the number of terminals.

## Contribution

It introduces an algorithm with complexity depending on the genus of the graph and proves a matching hardness result, closing the gap in complexity understanding.

## Key findings

- Algorithm with complexity f(R)·|V(G)|^{O(c_g·|T|)} for bounded genus graphs
- No f(R)·|V(G)|^{o(|T|^2/ log |T|)} algorithm exists for general graphs unless ETH fails
- Complexity depends critically on the genus and number of terminals

## Abstract

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.