Exact Bipartite Crossing Minimization under Tree Constraints

A

tanglegram

consists of a pair of (not necessarily binary) trees. Additional edges, called

tangles

, may connect the leaves of the first with those of the second tree. The task is to draw a tanglegram with a minimum number of tangle crossings while making sure that the trees are drawn crossing-free. This problem has relevant applications in computational biology, e.g., for the comparison of phylogenetic trees. Most existing approaches are only applicable for binary trees. In this work, we show that the problem can be formulated as a quadratic linear ordering problem (QLO) with side constraints. Buchheim et al. (INFORMS J. Computing, to appear) showed that, appropriately reformulated, the QLO polytope is a face of some cut polytope. It turns out that the additional side constraints do not destroy this property. Therefore, any polyhedral approach to max-cut can be used in our context. We present experimental results for drawing random and real-world tanglegrams defined on both binary and general trees. We evaluate linear as well as semidefinite programming techniques. By extensive experiments, we show that our approach is very efficient in practice.