On the complexity of covering points by disjoint segments and by guillotine cuts

2026-02-19

AI Summary

This study investigates the computational complexity of two planar geometric covering problems: determining whether a given point set can be completely covered by either k pairwise-disjoint line segments or k guillotine cuts. By presenting polynomial-time reductions from known NP-complete problems, the paper establishes for the first time that both covering problems are NP-complete in their general forms. This result precisely delineates the computational hardness boundary for these geometric covering tasks and provides a crucial theoretical foundation for the design of exact and approximation algorithms in subsequent research.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We show that two geometric cover problems in the plane, related to covering points with disjoint line segments, are NP-complete. Given $n$ points in the plane and a value $k$, the first problem asks if all points can be covered by $k$ disjoint line segments; the second problem treats the analogous question for $k$ guillotine cuts.

Problem

• geometric cover
• disjoint segments
• guillotine cuts
• NP-completeness
• point covering

Innovation

• NP-completeness
• geometric covering
• disjoint line segments
• guillotine cuts
• computational complexity