Parameterized Algorithms for the Steiner Arborescence Problem on a Hypercube

TL;DR

This paper develops efficient algorithms for the Steiner arborescence problem on directed hypercubes, exploiting hypercube structure and parameterized complexity to improve runtime and approximation quality.

Contribution

It introduces new algorithms with polynomial dependence on hypercube size and parameterized complexity, advancing solutions for the Steiner arborescence problem on hypercubes.

Findings

Exact algorithm with $ ilde{O}(3^{|R|})$ time.

Randomized algorithm with $ ilde{O}(9^q)$ time and high success probability.

Approximation algorithms with improved runtime and approximation guarantees.

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.