Parameterized Algorithms for the Steiner Arborescence Problem on a Hypercube
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2021
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| author | Mahapatra, Sugyani Narayanan, Manikandan Narayanaswamy, N S |
| author_facet | Mahapatra, Sugyani Narayanan, Manikandan Narayanaswamy, N S |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2110_02830 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Parameterized Algorithms for the Steiner Arborescence Problem on a Hypercube Mahapatra, Sugyani Narayanan, Manikandan Narayanaswamy, N S Data Structures and Algorithms 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. |
| title | Parameterized Algorithms for the Steiner Arborescence Problem on a Hypercube |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2110.02830 |