A Tighter Insertion-based Approximation of the Crossing Number

TL;DR

This paper introduces a polynomial-time algorithm for the multiple edge insertion problem that provides an additive approximation guarantee, improving bounds for crossing number approximations in F-almost-planar graphs.

Contribution

It presents the first directly implementable polynomial-time algorithm for MEI with an additive approximation guarantee based on F's size and G's maximum degree.

Findings

Achieves an additive approximation guarantee for MEI.

Provides improved crossing number bounds for F-almost-planar graphs.

Applicable to graphs with bounded degrees and small F.

Abstract

Let G be a planar graph and F a set of additional edges not yet in G. The multiple edge insertion problem (MEI) asks for a drawing of G+F with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. Finding an exact solution to MEI is NP-hard for general F. We present the first polynomial time algorithm for MEI that achieves an additive approximation guarantee -- depending only on the size of F and the maximum degree of G, in the case of connected G. Our algorithm seems to be the first directly implementable one in that realm, too, next to the single edge insertion. It is also known that an (even approximate) solution to the MEI problem would approximate the crossing number of the F-almost-planar graph G+F, while computing the crossing number of G+F exactly is NP-hard already when |F|=1. Hence our algorithm induces new, improved approximation bounds for the crossing number problem of F-almost-planar graphs, achieving constant-factor approximation for the large class of such graphs of bounded degrees and bounded size of F.