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.

Highlights

  • Domination problems in graphs are one of the most well-studied topics in theoretical computer science

  • We study the MIXED DOMINATING SET problem from the exact and parameterized viewpoint

  • We prove first that the problem can be solved in time O∗(5tw), and we prove that this algorithm and the one for pathwidth running in time O∗(5pw) (Jain et al (2017)) are optimal, up to polynomial factors, under the Strong Exponential Time Hypothesis (SETH)

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Summary

Introduction

Domination problems in graphs are one of the most well-studied topics in theoretical computer science. Jain et al showed that no algorithm can solve the problem in O∗((2−ε)pw) time under the Set Cover Conjecture (see Cygan et al (2012) for more details about the Set Cover Conjecture) These works observed that it is safe to assume that the optimal solution has a specific structure: the selected edges form a matching whose endpoints are disjoint from the set of selected vertices. This observation immediately gives an O∗(3n) algorithm for the problem, which was recently improved to O∗(2n) by Madathil et al (2019) by using a dynamic programming approach, which requires O∗(2n) space. (i) O∗ notation suppresses polynomial factors in the input size

Preliminaries
Treewidth
Exact Algorithm
FPT Algorithm
Conclusion
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