Is there FPT or XP algorithms knowm for Shortest Steiner cycle and $(a,b)$-Steiner path problem

Shortest Steiner cycle and $(a,b)$-Steiner path problem are generalizations of optimization versions of Hamiltonian cycle and Hamiltonian path problems.

The Shortest Steiner cycle problem is defined as follows:

Input: An undirected and unweighted graph $G=(V,E)$ and $T\subseteq V$ (called terminals).

Find: Shortest cycle (minimum edges) containing all terminals.

The Shortest $(a,b)-$Steiner path problem is defined as follows:

Input: An undirected and unweighted graph $G=(V,E)$, $T\subseteq V$ (called terminals) and two specific terminals $a,b \in T$.

Find: Shortest simple path (minimum edges) containing all terminals, which starts at $a$ and ends at $b$.

Are there XP (or even possibly FPT) algorithms known for these related problems, by parameter $k=|T|$?

PS: I could find only a randomized FPT algorithm for the Shortest Steiner cycle problem with runtime $2^{k}n^{\mathcal{O}(1)}$ in this paper.

Asked by user30486 (141 )
May 24, 2022 at 08:35