Parameterized algorithms for the Steiner arborescence problem on a hypercube - Mahapatra, Sugyani; Narayanan, Manikandan; Narayanaswamy, N. S. (Acta informatica, 2025)
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 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 Q→m comprising 2m 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≥0 or q≥0 as input, and outputs a Steiner arborescence with at most |R|+p-1 or m+q edges respectively. We present the following results (O~ hides the polynomial factors): An exact algorithm that runs in O~(3|R|) time.A randomized algorithm that runs in O~(9q) time with success probability ≥4-q.An exact algorithm that runs in O~(36q) time.A (1+q)-approximation algorithm that runs in O~(1.25284q) time.An Opℓmax-additive approximation algorithm that runs in O~(ℓmaxp+2) time, where ℓmax is the maximum distance of any terminal from the root.