Lower Bounds for Dominating Set in Ball Graphs and for Weighted Dominating Set in Unit-Ball Graphs

2020-05-10

Treewidth, Kernels, and Algorithms

AI Summary

This study investigates the computational complexity of the Dominating Set problem and its variants—such as Connected Dominating Set and Steiner Tree—in ball graphs and unit ball graphs. Under the Exponential Time Hypothesis (ETH), the authors establish fine-grained reductions to prove that Dominating Set in three-dimensional non-unit ball graphs and Weighted Dominating Set in unit ball graphs both admit no subexponential-time algorithms running in $2^{o(n)}$ time. This work presents the first $2^{o(n)}$-time lower bounds for several classical covering problems in these geometric graph models, highlighting a fundamental distinction in algorithmic tractability between unit and non-unit ball graphs and ruling out the existence of efficient subexponential algorithms for these problems.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

Recently it was shown that many classic graph problems—Independent Set, Dominating Set, Hamiltonian Cycle, and more—can be solved in subexponential time on unit-ball graphs. More precisely, these problems can be solved in \(2^{O(n^{1-1/d})}\) time on unit-ball graphs in \(\mathbb {R}^d\), which is tight under ETH. The result can be generalized to intersection graphs of similarly-sized fat objects.

Problem

• Dominating Set
• Unit-Ball Graphs
• Ball Graphs
• Weighted Dominating Set
• ETH

Innovation

• Dominating Set
• Unit-Ball Graphs
• ETH Lower Bounds
• Geometric Intersection Graphs
• Subexponential Algorithms