On Bipartite Drawings and the Linear Arrangement Problem

The bipartite crossing number problem is studied, and a connection between this problem and the linear arrangement problem is established. It is shown that when the arboricity is close to the minimum degree and the graph is not too sparse, then the optimal number of crossings has the same order of magnitude as the optimal arrangement value times the arboricity. The application of the results to a tree provides for a closed formula which expresses exactly, the optimal number of crossings in terms of the optimal value of the linear arrangement and the degree sequence, resulting in an O(n) time algorithm for computing the bipartite crossing number. Two polynomial time approximation algorithms for computing the bipartite crossing number are derived, with approximation factors, O(log n), and O(log n log logn), from the optimal, respectively, for approximating the number of crossings, and at the same time, total edge lengths, for a large class of graphs on n vertices. No approximation algorithm which could generate a provably good solution was previously known. The problem of computing a largest weighted biplanar subgraph of an acyclic graph is also studied and a linear time algorithm for it is derived. This problem was known to be NP-hard when the graph is planar and very sparse, and all weights are 1.