An Exact Approach to Upward Crossing Minimization

Abstract

The upward crossing number problem asks for a drawing of the graph into the plane with the minimum number of edge crossings where the edges are drawn as monotonously increasing curves w.r.t. the y-axis. While there is a large body of work on solving this central graph drawing problem heuristically, we present the first approach to solve the problem to proven optimality. Our approach is based on a reformulation of the problem as a boolean formula that can be iteratively tightened and resolved. In our experiments, we show the practical applicability and limits of our approach. Furthermore, we can now for the first time evaluate the state-of-the-art heuristics w.r.t. true optimum solutions. This leads to the finding that these algorithms are in general surprisingly far away from the optimum. Finally, we show that we can use our approach as a strong heuristic: even after only one minute of running time, our approach typically gives better solutions than the known heuristics for medium sized instances.

The Graph Crossing Number and its Variants: A Survey

“…Reference: Based on Eiglsperger, Kaufmann [198], also Chimani, Zeranski [139]. Comments: One of the monotone crossing numbers.…”

“…Comments: One of the monotone crossing numbers. The upward crossing number corresponds to the layer-free upward crossing minimization problem [134]. Eiglsperger and Kaufmann define the notion of a crossing number for a (mixed) upward planarization, calling it the (mixed) upward crossing minimal problem.…”

“…Eiglsperger and Kaufmann define the notion of a crossing number for a (mixed) upward planarization, calling it the (mixed) upward crossing minimal problem. Chimani and Zeranski [139] then use term upward crossing number. The upward crossing number could also be called the directed crossing number or the hierarchical crossing number; the latter term has been used in the context of leveled graphs [404].…”

“…Complexity: Even testing whether a graph is upward planar, that is, has upward crossing number 0, is NP-complete [246]. See [138] for a survey on upward planarity testing, and [139] for a survey on exact upward crossing minimization.…”