Finding Cliques in Geometric Intersection Graphs with Grounded or Stabbed Constraints

TL;DR

This paper investigates the computational complexity of finding maximum cliques in geometric intersection graphs with grounded or stabbed constraints, providing new hardness results, polynomial algorithms, and approximation methods for specific cases.

Contribution

It establishes NP-hardness for maximum clique problems in certain geometric intersection graphs and offers polynomial-time algorithms and approximation schemes for special cases.

Findings

NP-hardness for upward ray intersection graphs

Polynomial-time algorithm for grounded disks

3/2-approximation for disks with radii in [1,3]

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.