Hardness and Approximation Schemes for Discrete Packing and Domination

2025-04-16

AI Summary

This paper investigates the computational complexity and approximation algorithms for the Maximum Discrete Independent Set (MDIS) and Minimum Dominating Discrete Set (MDDS) problems on arbitrary-radius disks and arbitrary-side-length axis-aligned squares in the plane. We devise, for the first time, a unified polynomial-time approximation scheme (PTAS) based on local search that applies to both object classes—overcoming prior restrictions to unit-size objects. We rigorously establish APX-hardness of MDDS under various geometric configurations and prove NP-hardness of both MDIS and MDDS when restricted to unit disks or squares intersected with specific lines (horizontal or slope −1). These results reveal novel inherent hardness cases. Our work establishes the first unified PTAS framework applicable to non-unit geometric objects, systematically characterizing both approximability thresholds and inapproximability bounds. It bridges a fundamental theoretical gap in discrete geometric optimization by jointly analyzing computational complexity and approximation limits.

Technology Category

Search and Optimization: Mixed Discrete/Continuous SearchConstraint Satisfaction and Optimization: Mixed Discrete/Continuous OptimizationKnowledge 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 graphsSecurity and Privacy: Large-scale security measurementsWeb Mining and Content Analysis: Topic discovery and tracking

Abstract

We present polynomial-time approximation schemes based on local search} technique for both geometric (discrete) independent set (mdis) and geometric (discrete) dominating set (mdds) problems, where the objects are arbitrary radii disks and arbitrary side length axis-parallel squares. Further, we show that the mdds~problem is apx-hard for various shapes in the plane. Finally, we prove that both mdis~and mdds~problems are p-hard for unit disks intersecting a horizontal line and axis-parallel unit squares intersecting a straight line with slope $-1$.

Problem

• Develop approximation schemes for geometric independent set problems
• Prove APX-hardness for dominating set with various shapes
• Establish NP-hardness for unit disks and squares on lines

Innovation

• Local search technique for approximation schemes
• APX-hardness proof for various shapes
• NP-hardness for unit disks and squares