Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface

Abstract

The crossing number of a graph is the least number of pairwise edge crossings in a drawing of the graph in the plane. We provide an O(n log n) time constant factor approximation algorithm for the crossing number of a graph of bounded maximum degree which is "densely enough" embeddable in an arbitrary fixed orientable surface.Our approach combines some known tools with a powerful new lower bound on the crossing number of an embedded graph. This result extends previous results that gave such approximations in particular cases of projective, toroidal or apex graphs; it is a qualitative improvement over previously published algorithms that constructed low-crossing-number drawings of embeddable graphs without giving any approximation guarantees. No constant factor approximation algorithms for the crossing number problem over comparably rich classes of graphs are known to date.

A tighter insertion-based approximation of the crossing number

“…In [19], an algorithm is presented to approximate the crossing number of graphs embeddable in any fixed higher orientable surface. This algorithm lists the technical requirement that G has a "sufficiently dense" embedding on the surface.…”

“…This algorithm lists the technical requirement that G has a "sufficiently dense" embedding on the surface. Yet, as noted in [19], a result like Theorem 4.2 allows to drop this requirement: If the embedding density is small, then the removal of the offending small set(s) of edges is sufficient to reduce the graph genus, while the removed edges can be later inserted into an intermediate planar subgraph of the algorithm.…”

“…The known constant factor approximations restrict themselves to graphs following one of two paradigms (see also Section 4): they either assume that the graph is embeddable in some higher surface [14,19,21], or they are based on the idea that only a small set of graph elements has to be removed from G to make it planar: removing and re-inserting them can give strong approximation bounds [3,9,20]. In this paper, we follow the latter idea and first concentrate on the following tightly related problem: Definition 1.1 (Multiple edge insertion, MEI).…”

Crossing Numbers of Beyond-Planar Graphs

“…On the other hand, determining the crossing number of a graph is NP-hard [5] and can be solved exactly only on small/medium instances [6]. On the positive side, the crossing number is fixed-parameter tractable in the number of crossings [14] and can be approximated by a constant factor for graphs of bounded degree and genus [9].…”

Exact Crossing Number Parameterized by Vertex Cover

“…Concerning rectilinear drawings of dense graphs there is another recent approximation result [12]. Much better crossing number approximation results are known for some restricted graph classes, such as for graphs embeddable in a fixed surface [15,17] and for graphs from which few edges or vertices can be removed to make them planar [18,4,6,7].…”