Finding Cliques in Geometric Intersection Graphs with Grounded or Stabbed Constraints

2025-12-20

AI Summary

This paper investigates the Maximum Clique problem on geometric intersection graphs, focusing on two structural constraints: grounding (all objects intersect a common horizontal line) and piercing (a single line intersects all objects), for segments and disks. For upward-ray intersection graphs, we establish the first NP-hardness result for Maximum Clique. We design a polynomial-time algorithm for grounded unit-segment graphs, while proving Maximum Clique remains NP-hard for 1-pierced unit-segment graphs. We generalize the 3-pierceable property to disk families with radius ratio at most 3. For grounded disk graphs, we present a polynomial-time algorithm for Maximum Clique. Furthermore, for disk intersection graphs with radii in [1, 3], we devise a 3/2-approximation algorithm running in O(n³f(n)) time—improving upon existing EPTAS, QPTAS, and 2-approximation approaches.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• 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

Abstract

A geometric intersection graph is constructed over a set of geometric objects, where each vertex represents a distinct object and an edge connects two vertices if and only if the corresponding objects intersect. We examine the problem of finding a maximum clique in the intersection graphs of segments and disks under grounded and stabbed constraints. In the grounded setting, all objects lie above a common horizontal line and touch that line. In the stabbed setting, all objects can be stabbed with a common line. - We prove that finding a maximum clique is NP-hard for the intersection graphs of upward rays. This strengthens the previously known NP-hardness for ray graphs and settles the open question for the grounded segment graphs. The hardness result holds in the stabbed setting. - We show that the problem is polynomial-time solvable for intersection graphs of grounded unit-length segments, but NP-hard for stabbed unit-length segments. - We give a polynomial-time algorithm for the case of grounded disks. If the grounded constraint is relaxed, then we give an $O(n^3 f(n))$-time $3/2$-approximation for disk intersection graphs with radii in the interval $[1,3]$, where $n$ is the number of disks and $f(n)$ is the time to compute a maximum clique in an $n$-vertex cobipartite graph. This is faster than previously known randomized EPTAS, QPTAS, or 2-approximation algorithms for arbitrary disks. We obtain our result by proving that pairwise intersecting disks with radii in $[1,3]$ are 3-pierceable, which extends the 3-pierceable property from the long known unit disk case to a broader class.

Problem

• Finding maximum cliques in geometric intersection graphs with constraints.
• Determining computational complexity for grounded and stabbed segment graphs.
• Developing algorithms for disk intersection graphs with radius constraints.

Innovation

• NP-hardness proof for upward rays intersection graphs
• Polynomial-time algorithm for grounded unit-length segments
• 3-pierceable property for disks with radii in [1,3]