Exploring Periodic Dominating Sets in Bounded-Treewidth Graphs

Core Concepts

The author investigates the complexity of (𝜎, 𝜌)-DomSet for periodic sets, providing optimal algorithms and matching lower bounds under SETH.

Abstract

The content delves into the study of generalized dominating set problems, focusing on periodic sets with the same period. It presents an algorithm that optimally solves decision, minimization, and maximization versions of the problem. The analysis includes upper bounds based on dynamic programming and convolution techniques, as well as lower bounds derived from reductions to 𝑞-CSP-𝐵. The research contributes to understanding the intricacies of solving dominating set problems efficiently in bounded-treewidth graphs.

Stats

We extend the recent result by Focke et al. [SODA 2023] to investigate the case when 𝜎 and 𝜌 are periodic sets with the same period m ≥ 2.
We present an algorithm that solves in time mtw · 𝑛𝒪(1) the decision, minimization and maximization version of the problem.
For all difficult cases which are covered by our algorithm we show a lower bound for the decision version indicating that there is no (m − 𝜀)pw · 𝑛𝒪(1) unless SETH fails.

Quotes

"The clear goal is to identify the smallest value for 𝑐 such that there is no (𝑐 − 𝜀)tw · 𝑛𝒪(1) algorithm under the Strong Exponential-Time Hypothesis (SETH)." - J. Greilhuber, P. Schepper, and P. Wellnitz

Key Insights Distilled From

Shining Light on Periodic Dominating Sets in Bounded-Treewidth Graphs

by Jakob Greilh... at arxiv.org 03-13-2024

https://arxiv.org/pdf/2403.07524.pdf

Deeper Inquiries

What implications do these findings have for other vertex selection problems

The findings in the research on periodic dominating sets have significant implications for other vertex selection problems. By extending the analysis to cover periodic sets with the same period, the study provides a framework for understanding and solving generalized dominating set problems efficiently. This approach can be applied to various related vertex selection problems that involve degree constraints, such as Independent Set, Perfect Code, Total Dominating Set, and more. The algorithmic techniques developed in this research can be adapted and utilized to improve algorithms for these related problems.

How do periodic sets with different periods affect the complexity of Dominating Set problems

When considering periodic sets with different periods in Dominating Set problems, the complexity of the problem increases significantly. Unlike when dealing with sets of integers that share a common period m ≥ 2, where optimal algorithms can be designed within mtw · 𝑛𝒪(1) time complexity, having different periods introduces additional challenges. The structural differences between sets with distinct periods may require novel algorithmic approaches tailored to handle these complexities effectively. Therefore, analyzing Dominating Set problems involving periodic sets with varying periods would likely lead to more intricate algorithm designs and potentially higher computational complexities.

Can insights from this research be applied to optimize algorithms for other combinatorial optimization problems

Insights from this research on optimizing algorithms for periodic dominating set problems can indeed be applied to enhance solutions for other combinatorial optimization problems as well. By leveraging techniques like dynamic programming on tree decompositions and efficient state compression methods based on structural properties of integer sets (such as sparseness), similar optimization strategies could be implemented in various combinatorial optimization contexts. Problems requiring constraint satisfaction or variable assignments could benefit from adopting similar approaches to reduce search space size and improve overall efficiency in finding solutions within specific constraints or conditions.