The cut polytope and the Boolean quadric polytope
Discrete Mathematics
A Branch and Bound Algorithm for Max-Cut Based on Combining Semidefinite and Polyhedral Relaxations
IPCO '07 Proceedings of the 12th international conference on Integer Programming and Combinatorial Optimization
Untangling Tanglegrams: Comparing Trees by Their Drawings
ISBRA '09 Proceedings of the 5th International Symposium on Bioinformatics Research and Applications
A Faster Fixed-Parameter Approach to Drawing Binary Tanglegrams
Parameterized and Exact Computation
Exact Algorithms for the Quadratic Linear Ordering Problem
INFORMS Journal on Computing
Comparing trees via crossing minimization
Journal of Computer and System Sciences
Generalized k-ary tanglegrams on level graphs: A satisfiability-based approach and its evaluation
Discrete Applied Mathematics
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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.