On 3-layer crossings and pseudo arrangements

Abstract

Let G = (V0; V1; V2;E) be a 3-layer graph. The 3-layer drawings of G in which V0, V1, and V2 are placed on 3 parallel lines and each edge in E is drawn using one straight line segment, are studied. A generalization of the linear arrangement problem which we call the 3-layer pseudo linear arrangement problem is introduced, and it is shown to be closely related to the 3-layer crossing number. In particular, we show that the 3-layer crossing number of G plus the sum of the square of degrees asymptotically has the same order of magnitude as the opti-mal solution to the 3-layer linear arrangement problem. Consequently, when G satisfies certain (reasonable) assumptions, we derive the first po-lynomial time approximation algorithm to compute the 3-layer crossing number within a multiplicative factor of O(log n) from the optimal.