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 (qge 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 (ge 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( pell _{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.