Approximating the Bundled Crossing Number

TL;DR

This paper presents a polynomial-time algorithm that approximates the bundled crossing number in good graph drawings within a factor of 8, improving previous bounds and extending to pseudosegment families.

Contribution

It introduces a new 8-approximation algorithm for the bundled crossing number in good drawings, including special cases like circular drawings and pseudosegments.

Findings

Polynomial-time 8-approximation algorithm for bundled crossing number.

Improved approximation factor for circular drawings from 10 to 8.

Extension of the approach to pseudosegment families and bipartite intersection graphs.

Abstract

Bundling crossings is a strategy which can enhance the readability of drawings. In this paper we consider good drawings, i.e., we require that any two edges have at most one common point which can be a common vertex or a crossing. Our main result is that there is a polynomial time algorithm to compute an 8-approximation of the bundled crossing number of a good drawing (up to adding a term depending on the facial structure of the drawing). In the special case of circular drawings the approximation factor is 8 (no extra term), this improves upon the 10-approximation of Fink et al. (Bundled crossings in embedded graphs, Proc. Latin'16). Our approach also works with the same approximation factor for families of pseudosegments, i.e., curves intersecting at most once. We also show how to compute a 9/2-approximation when the intersection graph of the pseudosegments is bipartite.