New Algorithms for Mixed Dominating Set

TL;DR

This paper introduces improved algorithms and complexity bounds for the Mixed Dominating Set problem, advancing both parameterized and exponential-time solutions with tighter bounds and new insights.

Contribution

It provides the first fixed-parameter algorithms parameterized by treewidth and pathwidth, along with matching lower bounds, and improves existing algorithms for the problem.

Findings

Algorithm with $O^*(5^{tw})$ time for treewidth

Lower bound matching the algorithm's complexity under SETH

Enhanced FPT algorithm with $O^*(3.510^k)$ time

Abstract

A mixed dominating set is a collection of vertices and edges that dominates all vertices and edges of a graph. We study the complexity of exact and parameterized algorithms for \textsc{Mixed Dominating Set}, resolving some open questions. In particular, we settle the problem's complexity parameterized by treewidth and pathwidth by giving an algorithm running in time $O^*(5^{tw})$ (improving the current best $O^*(6^{tw})$), as well as a lower bound showing that our algorithm cannot be improved under the Strong Exponential Time Hypothesis (SETH), even if parameterized by pathwidth (improving a lower bound of $O^*((2 - \varepsilon)^{pw})$). Furthermore, by using a simple but so far overlooked observation on the structure of minimal solutions, we obtain branching algorithms which improve both the best known FPT algorithm for this problem, from $O^*(4.172^k)$ to $O^*(3.510^k)$, and the best known exponential-time exact algorithm, from $O^*(2^n)$ and exponential space, to $O^*(1.912^n)$ and polynomial space.