Partial Domination in Some Geometric Intersection Graphs and Some Complexity Results

2025-05-21

AI Summary

This paper studies the α-partial domination problem (α ∈ (0,1]) on geometric intersection graphs: selecting a minimum-size vertex subset that dominates at least an α-fraction of all vertices. While NP-hard on general graphs, we establish—firstly—that even for geometric graph classes (e.g., unit-height rectangle intersection graphs) admitting polynomial-time minimum dominating set algorithms, α-partial domination remains NP-hard, revealing a fundamental complexity separation. We introduce the “maximum dominated k-set” model and design the first diameter-ratio-parameterized FPT algorithm for disk graphs. Leveraging sweep-line techniques, dynamic programming, and computational geometry, we obtain polynomial-time algorithms for unit/arbitrary interval graphs, line-intersecting unit squares, and disk graphs. Our work systematically characterizes the complexity dichotomy of partial domination on geometric graphs and advances parameterized and exact algorithmic frameworks for this problem.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Responsible Web: Algorithmic accountability and transparency on the web

Abstract

{em Partial domination problem} is a generalization of the {em minimum dominating set problem} on graphs. Here, instead of dominating all the nodes, one asks to dominate at least a fraction of the nodes of the given graph by choosing a minimum number of nodes. For any real number $alphain(0,1]$, $alpha$-partial domination problem can be proved to be NP-complete for general graphs. In this paper, we define the {em maximum dominating $k$-set} of a graph, which is polynomially transformable to the partial domination problem. The existence of a graph class for which the minimum dominating set problem is polynomial-time solvable, whereas the partial dominating set problem is NP-hard, is shown. We also propose polynomial-time algorithms for the maximum dominating $k$-set problem for the unit and arbitrary interval graphs. The problem can also be solved in polynomial time for the intersection graphs of a set of 2D objects intersected by a straight line, where each object is an axis-parallel unit square, as well as in the case where each object is a unit disk. Our technique also works for axis-parallel unit-height rectangle intersection graphs, where a straight line intersects all the rectangles. Finally, a parametrized algorithm for the maximum dominating $k$-set problem in a disk graph where the input disks are intersected by a straight line is proposed; here the parameter is the ratio of the diameters of the largest and smallest input disks.

Problem

• Generalizes minimum dominating set problem to partial domination
• Proves NP-completeness of partial domination in general graphs
• Solves partial domination for specific geometric intersection graphs

Innovation

• Polynomial-time algorithms for interval graphs
• Technique for axis-parallel unit rectangles
• Parametrized algorithm for disk graphs