Structural Parameterizations of Dominating Set Variants

Abstract

We consider structural parameterizations of the fundamental dominating set problem and its variants in the parameter ecology program. We give improved fixed-parameter tractable (FPT) algorithms and lower bounds under well-known conjectures for dominating set in graphs that are k vertices away from a cluster graph or a split graph. These are graphs in which there is a set of k vertices (called the modulator) whose deletion results in a cluster graph or a split graph. We also call k as the deletion distance (to the appropriate class of graphs). Specifically, we show the following results. When parameterized by the deletion distance k to cluster graphs,

• we can find a minimum dominating set in ${\mathcal O}^*(3^k)$ time (${\mathcal O}^*$ notation ignores polynomial factors of input). Within the same time, we can also find a minimum independent dominating set (IDS) or a minimum efficient dominating set (EDS) or a minimum total dominating set. These algorithms are obtained through a dynamic programming approach for an interesting generalization of set cover which may be of independent interest.

• We complement our upper bound results by showing that at least for dominating set and total dominating set, ${\mathcal O}^*((2-\epsilon )^k)$ time algorithm is not possible for any $\epsilon > 0$ under, what is known as, Set Cover Conjecture. We also show that most of these variants of dominating set do not have polynomial sized kernel.

The standard dominating set and most of its variants are $\mathsf {NP}$-hard or $\mathsf {W}$[2]-hard in split graphs. For the two variants IDS and EDS that are polynomial time solvable in split graphs, we show that when parameterized by the deletion distance k to split graphs,

• IDS can be solved in ${\mathcal O}^*(2^k)$ time and we provide an $\Omega (2^k)$ lower bound under the strong exponential time hypothesis (SETH);

• the $2^k$ barrier can be broken for EDS by designing an ${\mathcal O}^*( 3^{k/2})$ algorithm. This is one of the very few problems with a runtime better than ${\mathcal O}^*(2^k)$ in the realm of structural parameterization. We also show that no $2^{o(k)}$ algorithm is possible unless the exponential time hypothesis (ETH) is false.