Pathways to Tractability for Geometric Thickness

TL;DR

This paper explores the computational complexity of geometric thickness, proposing fixed-parameter algorithms based on graph parameters and extending partial solutions to find tractability pathways.

Contribution

It introduces fixed-parameter algorithms using vertex cover and feedback edge parameters and characterizes complexity in extension settings based on missing elements.

Findings

Fixed-parameter algorithms for geometric thickness

Characterization of complexity in extension problems

Pathways to tractability via partial solution extensions

Abstract

We study the classical problem of computing geometric thickness, i.e., finding a straight-line drawing of an input graph and a partition of its edges into as few parts as possible so that each part is crossing-free. Since the problem is NP-hard, we investigate its tractability through the lens of parameterized complexity. As our first set of contributions, we provide two fixed-parameter algorithms which utilize well-studied parameters of the input graph, notably the vertex cover and feedback edge numbers. Since parameterizing by the thickness itself does not yield tractability and the use of other structural parameters remains open due to general challenges identified in previous works, as our second set of contributions, we propose a different pathway to tractability for the problem: extension of partial solutions. In particular, we establish a full characterization of the problem's parameterized complexity in the extension setting depending on whether we parameterize by the number of missing vertices, edges, or both.