Parameterized algorithms for the Steiner arborescence problem on a hypercube

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 FPT time. We explore the MSA-DH problem on three natural parameters—\(|R|\), and two above-guarantee parameters, number of Steiner nodes p and penalty q (defined as the extra cost above m incurred by the solution). For above-guarantee parameters, the parameterized MSA-DH problem take \(p \ge 0\) or \(q\ge 0\) 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. 1.

   An exact algorithm that runs in \(\tilde{{\mathcal {O}}}(3^{|R|})\) time.

2. 2.

   A randomized algorithm that runs in \(\tilde{{\mathcal {O}}}(9^q)\) time with success probability \(\ge 4^{-q}\).

3. 3.

   An exact algorithm that runs in \(\tilde{{\mathcal {O}}}(36^q)\) time.

4. 4.

   A \((1+q)\)-approximation algorithm that runs in \(\tilde{{\mathcal {O}}}(1.25284^q)\) time.

5. 5.

   An \({\mathcal {O}}\left( p\ell _{\textrm{max}}\right) \)-additive approximation algorithm that runs in \(\tilde{{\mathcal {O}}}(\ell _{\textrm{max}}^{p+2})\) time, where \(\ell _{\textrm{max}}\) is the maximum distance of any terminal from the root.