Finding a Maximum Clique in a Grounded 1-Bend String Graph

TL;DR

This paper investigates the computational complexity of finding maximum cliques in grounded 1-bend string graphs, proving APX-hardness in various cases and providing efficient algorithms for specific subclasses.

Contribution

It establishes APX-hardness for maximum clique problems in grounded 1-bend string graphs and introduces fast algorithms for certain subclasses.

Findings

Maximum clique problem is APX-hard for grounded 1-bend string graphs.

APX-hardness persists even with restricted bend and endpoint positions.

Efficient algorithms are developed for subclasses of grounded segment graphs.

Abstract

A grounded 1-bend string graph is an intersection graph of a set of polygonal lines, each with one bend, such that the lines lie above a common horizontal line $\ell$ and have exactly one endpoint on $\ell$. We show that the problem of finding a maximum clique in a grounded 1-bend string graph is APX-hard, even for strictly $y$-monotone strings. For general 1-bend strings, the problem remains APX-hard even if we restrict the position of the bends and end-points to lie on at most three parallel horizontal lines. We give fast algorithms to compute a maximum clique for different subclasses of grounded segment graphs, which are formed by restricting the strings to various forms of $L$-shapes.