Parameterized Algorithms for the Steiner Arborescence Problem on a Hypercube

Abstract

Motivated by a phylogeny reconstruction problem in evolutionary biology, we study the minimum Steiner arborescence problem on directed hypercubes (MSA-DH). Given $m$, representing the directed hypercube $\vec{Q}_m$, and a set of terminals $R$, the problem asks to find a Steiner arborescence that spans $R$ with minimum cost. As $m$ implicitly represents $\vec{Q}_m$ comprising $2^{m}$ vertices, the running time analyses of traditional Steiner tree algorithms on general graphs does not give a clear understanding of the actual complexity of this problem. We present algorithms that exploit the structure of the hypercube and run in time polynomial in $|R|$ and $m$. We explore the MSA-DH problem on three natural parameters - $R$, and two above-guarantee parameters, number of Steiner nodes $p$ and penalty $q$. For above-guarantee parameters, the parameterized MSA-DH problem takes $p \geq 0$ or $q\geq 0$ as input, and outputs a Steiner arborescence with at most $|R| + p - 1$ or $m + q$ edges respectively. We present the following results ($\tilde{\mathcal{O}}$ hides the polynomial factors): 1. An exact algorithm that runs in $\tilde{\mathcal{O}}(3^{|R|})$ time. 2. A randomized algorithm that runs in $\tilde{\mathcal{O}}(9^q)$ time with success probability $\geq 4^{-q}$. 3. An exact algorithm that runs in $\tilde{\mathcal{O}}(36^q)$ time. 4. A $(1+q)$-approximation algorithm that runs in $\tilde{\mathcal{O}}(1.25284^q)$ time. 5. An $\mathcal{O}\left(p\ell_{\mathrm{max}} \right)$-additive approximation algorithm that runs in $\tilde{\mathcal{O}}(\ell_{\mathrm{max}}^{p+2})$ time, where $\ell_{\mathrm{max}}$ is the maximum distance of any terminal from the root.