A subpolynomial approximation algorithm for graph crossing number in low-degree graphs

Abstract

We consider the classical Minimum Crossing Number problem: given an n-vertex graph G, compute a drawing of G in the plane, while minimizing the number of crossings between the images of its edges. This is a fundamental and extensively studied problem, whose approximability status is widely open. In all currently known approximation algorithms, the approximation factor depends polynomially on Δ – the maximum vertex degree in G. The best current approximation algorithm achieves an O(n1/2−· (Δ·logn))-approximation, for a small fixed constant є, while the best negative result is APX-hardness, leaving a large gap in our understanding of this basic problem. In this paper we design a randomized O(2O((logn)7/8loglogn)·(Δ))-approximation algorithm for Minimum Crossing Number. This is the first approximation algorithm for the problem that achieves a subpolynomial in n approximation factor (albeit only in graphs whose maximum vertex degree is subpolynomial in n).

Locations

• arXiv (Cornell University) - View - PDF