Parameterized and Approximation Algorithms for the Steiner Arborescence Problem on a Hypercube

We address the Steiner arborescence problem on a directed hypercube. The directed hypercube enjoys a special connectivity structure among its node set $\{0,1\}^m$, but its exponential in $m$ size renders traditional Steiner tree algorithms inefficient. Even though the problem was known to be NP-hard, parameterized complexity or approximation algorithms for the problem was unknown. With applications in evolutionary/phylogenetic tree reconstruction algorithms and incremental algorithms for computing a property on multiple input graphs, any algorithm for this problem would open up new ways to study these applications. Given the dimension $m$ of a directed hypercube $\vec{Q}_m$ defined over the node set $V(\vec{Q}_m) = \{0,1\}^m$, and a set of terminals $R \subseteq V(\vec{Q}_m)$, the \textsc{Minimum Steiner Arborescence} (MSA) problem seeks to find a Steiner arborescence $T$ rooted at $0^m$ that spans $R$ with the minimum cost. We study this problem with respect to three different parameters: (i) the number of terminals $|R|$, (ii) the optimal number of Steiner nodes ($p_{opt}$), and (iii) the penalty $q := \mathtt{COST}(T) - m$ (while $q_{opt}$ denotes the optimal penalty). In this article, we present the following parameterized and approximation results. 1. The MSA problem admits an $\mathsf{FPT}$ (\emph{fixed-parameter tractable}) algorithm parameterized by $|R|$. 2. The MSA problem admits an $\mathsf{FPT}$ algorithm parameterized by $q$. 3. The MSA problem admits a $(1+2q_{opt})$-approximation and a $\min\{(\ell-1), \frac{p_{opt}}{2}\} \times \min\{\ell, (\ln |R| + 2)\}$-approximation algorithms where $\ell$ is the maximum distance of any terminal from the root $0^m$.