Parameterized Complexity of Weighted Multicut in Trees

Parameterized algorithms for Steiner tree and (connected) dominating set on path graphs

“…Hence, Theorem 6 follows from the above lemma. Additionally, it is known that Dominating Set can be solved in time $$$ {2}^{O\left(\ell \right)}\cdotp {n}^{O(1)} $$$ on a chordal graph $$$ G $$$, where $$$ \ell =\ell (G) $$$ and $$$ n=\mid V(G)\mid $$$ [16]. Since the hypothesis of Lemma 2 applies to Dominating Set , we get that it can be solved in time $$$ {2}^{O\left(\kappa \cdotp v\mathit{\ell}(G)\right)}\cdotp {n}^{O(1)} $$$ on a chordal graph, provided the appropriate model is given, that is, the part of Theorem 2 concerning Dominating Set follows.…”

“…We now prove the part of this theorem concerning Dominating Set and the remainder follows from Theorem 3, which we prove in Sections 5 and 6. Even though this complexity is worse than the one achieved by Theorem 2, we propose this algorithm for its simplicity when compared to the one presented in [16] that we used to conclude Theorem 2.…”

“…The present full paper extends the preliminary results published at WALCOM 2022 [12] by establishing a hardness relation between Dominating Set , Connected Dominating Set and Steiner tree , including the weighted versions of the problems and considering complexity separating subclasses of chordal graphs. An independent related work has recently been presented at IPEC 2022 [16], where they study the complexity of these problems for the class of chordal graphs when parameterized by the leafage, improving results presented in [15].…”

“…We nevertheless present our slightly slower algorithm because, being designed specifically to undirected path graphs, it is much simpler than the one in [16]. In Theorem 2 below, however, we apply the algorithm in [16] directly.…”

“…This unfortunately is not the case for the vertex leafage parameter, as it is known [9] that it is $$$ \mathrm{NP} $$$‐complete to decide whether a chordal graph $$$ G $$$ has vertex leafage at most 3; they also give an algorithm to compute $$$ v\mathit{\ell}(G) $$$ in time $$$ {n}^{\ell (G)} $$$, which is $$$ \mathrm{XP} $$$ when parameterized by $$$ \ell (G) $$$. In [15] they provide an $$$ \mathrm{FPT} $$$ algorithm for Dominating Set when parameterized by $$$ \ell (G) $$$, which has been recently improved in [16]. Since $$$ v\mathit{\ell}(G)\le \ell (G) $$$, we get that $$$ v\mathit{\ell}(G) $$$ is a more restrictive parameter than $$$ \ell (G) $$$, and hence these algorithms are not readily applicable to Dominating Set parameterized by $$$ \kappa $$$ and $$$ v\mathit{\ell}(G) $$$.…”