Given an n-vertex graph G, a drawing of G in the plane is a mapping of its vertices into points of the plane, and its edges into continuous curves, connecting the images of their endpoints. A crossing in such a drawing is a point where two such curves intersect. In the Minimum Crossing Number problem, the goal is to find a drawing of G with minimum number of crossings. The value of the optimal solution, denoted by OPT, is called the graph's crossing number. This is a very basic problem in topological graph theory, that has received a significant amount of attention, but is still poorly understood algorithmically. The best currently known efficient algorithm produces drawings with @math crossings on bounded-degree graphs, while only a constant factor hardness of approximation is known. A closely related problem is Minimum Edge Planarization, in which the goal is to remove a minimum-cardinality subset of edges from G, such that the remaining graph is planar. Our main technical result establishes the following connection between the two problems: if we are given a solution of cost k to the Minimum Edge Planarization problem on graph G, then we can efficiently find a drawing of G with at most @math crossings, where @math is the maximum degree in G. This result implies an @math -approximation for Minimum Crossing Number, as well as improved algorithms for special cases of the problem, such as, for example, k-apex and bounded-genus graphs.
Grohe @cite_35 , answering a question of Downey and Fellows @cite_0 , proved that the crossing number is fixed-parameter tractable. In particular, for any fixed number of crossings his algorithm computes an optimal drawing in @math time. Building upon the breakthrough result of Mohar @cite_22 for embedding graphs into a surface of bounded genus, Kawarabayashi and Reed @cite_10 gave an improved fixed-parameter algorithm with running time @math .
@cite_0 @cite_35 @cite_10 @cite_22
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We show that for every fixed k, there is a linear time algorithm that decides whether or not a given graph has crossing number at most k, and if this is the case, computes a drawing of the graph in the plane with at most k crossings. This answers the question posed by Grohe (STOC'01 and JCSS 2004). Our algorithm can be viewed as a generalization of the seminal result by Hopcroft and Tarjan lin1, which determines if a given graph is planar in linear time. Our algorithm can also be compared to the algorithms by Mohar (STOC'96 and Siam J. Discrete Math 2001), for testing the embeddability of an input graph in a fixed surface. For each surface s, Mohar describes an algorithm which yields either an embedding of G in s or a minor of G which is not embeddable in s and is minimal with this property. The same approach allows us to obtain linear time algorithms for the same question for a variety of other crossing numbers. We can also determine in linear time if an input graph can be made planar by the deletion of k edges (for fixed k). We show that for each integer g 0 there is a constant cg > 0 such that every graph that embeds in the projective plane with sucien tly large face width r has crossing number at least cgr 2 in the orientable surface g of genus g. As a corollary, we give a polynomial time constant factor approximation algorithm for the crossing number of projective graphs with bounded degree. In this paper we study the problem of computing subgraphs of a certain configuration in a given topological graph G such that the number of crossings in the subgraph is minimum. The configurations that we consider are spanning trees, s-t paths, cycles, matchings, and @k-factors for @[email protected]? 1,2 . We show that it is NP-hard to approximate the minimum number of crossings for these configurations within a factor of k^1^-^@e for any @e>0, where k is the number of crossings in G. We then give a simple fixed-parameter algorithm that tests in O^@?(2^k) time whether G has a crossing-free configuration for any of the above, where the O^@?-notation neglects polynomial terms. For some configurations we have faster algorithms. The respective running times are O^@?(1.9999992^k) for spanning trees and O^@?((3)^k) for s-t paths and cycles. For spanning trees we also have an O^@?(1.968^k)-time Monte-Carlo algorithm. Each O^@?(@b^k)-time decision algorithm can be turned into an O^@?((@b+1)^k)-time optimization algorithm that computes a configuration with the minimum number of crossings. It was proved by [M.R. Garey, D.S. Johnson, Crossing number is NP-complete, SIAM J. Algebraic Discrete Methods 4 (1983) 312-316] that computing the crossing number of a graph is an NP-hard problem. Their reduction, however, used parallel edges and vertices of very high degrees. We prove here that it is NP-hard to determine the crossing number of a simple 3-connected cubic graph. In particular, this implies that the minor-monotone version of the crossing number problem is also NP-hard, which has been open till now.