Drawing graphs with mathematical programming and variable neighborhood search

In the Graph Drawing problem a set of symbols must be placed in a plane and their connections routed. To produce clear, easy to read diagrams, this problem is usually solved trying to minimize edges crossing and the area occupied, resulting in a NP-Hard problem. Our research focuses in drawing Entity Relationship (ER) diagrams, a challenging version of the problem where nodes have different sizes. Mathematical Programming models for the two solution phases, node placement and connection routing, are discussed and their exact resolution by an Integer Programming (IP) solver is evaluated. As the first phase proved to be specially hard to be solved exactly, a hybrid Variable Neighborhood Search (VNS) heuristic is proposed. Using IP techniques we obtained provably optimal (or close to optimal) solutions for the two different phases, at the expense of a large computational effort. We also show that our VNS based heuristic approach can produce close to optimal solutions in very short times for the hardest part of the solution process. Using either methods we have produced clearly better drawings than existing solutions.