An Improved Neural Network Model for the Two-Page Crossing Number Problem

The simplest graph drawing method is that of putting the vertices of a graph on a line and drawing the edges as half-circles either above or below the line. Such drawings are called two-page book drawings. The smallest number of crossings over all two-page drawings of a graph <emphasis><formula formulatype="inline"> <tex>$G$</tex></formula></emphasis> is called the two-page crossing number of <emphasis><formula formulatype="inline"><tex>$G$</tex></formula></emphasis>. Cimikowski and Shope have solved the two-page crossing number problem for an <emphasis><formula formulatype="inline"><tex>$n$</tex></formula></emphasis>-vertex and <emphasis><formula formulatype="inline"><tex>$m$</tex></formula></emphasis>-edge graph by using a Hopfield network with 2 <emphasis><formula formulatype="inline"> <tex>$m$</tex></formula></emphasis> neurons. We present here an improved Hopfield model with <emphasis><formula formulatype="inline"><tex>$m$</tex></formula></emphasis> neurons. The new model achieves much better performance in the quality of solutions and is more efficient than the model of Cimikowski and Shope for all graphs tested. The parallel time complexity of the algorithm, without considering the crossing number calculations, is <emphasis><formula formulatype="inline"> <tex>$O(m)$</tex></formula></emphasis> for the new Hopfield model with <emphasis><formula formulatype="inline"><tex>$m$</tex></formula></emphasis> processors clearly outperforming the previous algorithm.