Exact Bipartite Crossing Minimization under Tree Constraints

A tanglegram consists of a pair of (not necessarily binary) trees T_1, T_2 with leaf sets L_1, L_2. Additional edges, called tangles, may connect nodes in L_1 with those in L_2. The task is to draw the tanglegram with a minimum number of tangle edge crossings while making sure that no crossing occurs between edges within each tree. This problem has relevant applications in computational biology, e.g., for the comparison of phylogenetic trees. In this work, we show that the problem can be formulated as a quadratic linear ordering problem (QLO) with additional side constraints. It was already shown that, appropriately reformulated, the QLO polytope is a face of some cut polytope. It turns out that the additional side constraints arising in our application 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 realistic tanglegrams using both linear and semidefinite programming techniques, showing that our approach is very efficient in practice

Exact Bipartite Crossing Minimization under Tree Constraints

A tanglegram consists of a pair of (not necessarily binary) trees T_1, T_2 with leaf sets L_1, L_2. Additional edges, called tangles, may connect nodes in L_1 with those in L_2. The task is to draw the tanglegram with a minimum number of tangle edge crossings while making sure that no crossing occurs between edges within each tree. This problem has relevant applications in computational biology, e.g., for the comparison of phylogenetic trees. In this work, we show that the problem can be formulated as a quadratic linear ordering problem (QLO) with additional side constraints. It was already shown that, appropriately reformulated, the QLO polytope is a face of some cut polytope. It turns out that the additional side constraints arising in our application 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 realistic tanglegrams using both linear and semidefinite programming techniques, showing that our approach is very efficient in practice

The largest crossing number of tanglegrams

A tanglegram $\cal T$ consists of two rooted binary trees with the same number of leaves, and a perfect matching between the two leaf sets. In a layout, the tanglegrams is drawn with the leaves on two parallel lines, the trees on either side of the strip created by these lines are drawn as plane trees, and the perfect matching is drawn in straight line segments inside the strip. The tanglegram crossing number ${\rm cr}({\cal T})$ of $\cal T$ is the smallest number of crossings of pairs of matching edges, over all possible layouts of $\cal T$. The size of the tanglegram is the number of matching edges, say $n$. An earlier paper showed that the maximum of the tanglegram crossing number of size $n$ tanglegrams is $<\frac{1}{2}\binom{n}{2}$; but is at least $\frac{1}{2}\binom{n}{2}-\frac{n^{3/2}-n}{2}$ for infinitely many $n$. Now we make better bounds: the maximum crossing number of a size $n$ tanglegram is at most $\frac{1}{2}\binom{n}{2}-\frac{n}{4}$, but for infinitely many $n$, at least $\frac{1}{2}\binom{n}{2}-\frac{n\log_2 n}{4}$. The problem shows analogy with the Unbalancing Lights Problem of Gale and Berlekamp

Visualizing Co-Phylogenetic Reconciliations

We introduce a hybrid metaphor for the visualization of the reconciliations of co-phylogenetic trees, that are mappings among the nodes of two trees. The typical application is the visualization of the co-evolution of hosts and parasites in biology. Our strategy combines a space-filling and a node-link approach. Differently from traditional methods, it guarantees an unambiguous and `downward' representation whenever the reconciliation is time-consistent (i.e., meaningful). We address the problem of the minimization of the number of crossings in the representation, by giving a characterization of planar instances and by establishing the complexity of the problem. Finally, we propose heuristics for computing representations with few crossings.Comment: This paper appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Tanglegrams are misleading for visual evaluation of tree congruence

Evolutionary Biologists are often faced with the need to compare phylogenetic trees. One popular method consists in visualizing the trees face to face with links connecting matching taxa. These tanglegrams are optimized beforehand so that the number of lines crossing (the entanglement) is minimal. This representation is implicitly justified by the expectation that the level of entanglement is correlated with the level of similarity (or congruence) between the trees compared. Using simulations, we show that this correlation is actually very weak, which should preclude the use of such technique for getting insight into the level of congruence between trees

Exact and fixed-parameter algorithms for metro-line crossing minimization problems

A metro-line crossing minimization problem is to draw multiple lines on an underlying graph that models stations and rail tracks so that the number of crossings of lines becomes minimum. It has several variations by adding restrictions on how lines are drawn. Among those, there is one with a restriction that line terminals have to be drawn at a verge of a station, and it is known to be NP-hard even when underlying graphs are paths. This paper studies the problem in this setting, and propose new exact algorithms. We first show that a problem to decide if lines can be drawn without crossings is solved in polynomial time, and propose a fast exponential algorithm to solve a crossing minimization problem. We then propose a fixed-parameter algorithm with respect to the multiplicity of lines, which implies that the problem is FPT.Comment: 19 pages, 15 figure

Randomized approximation algorithms : facility location, phylogenetic networks, Nash equilibria

Despite a great effort, researchers are unable to find efficient algorithms for a number of natural computational problems. Typically, it is possible to emphasize the hardness of such problems by proving that they are at least as hard as a number of other problems. In the language of computational complexity it means proving that the problem is complete for a certain class of problems. For optimization problems, we may consider to relax the requirement of the outcome to be optimal and accept an approximate (i.e., close to optimal) solution. For many of the problems that are hard to solve optimally, it is actually possible to efficiently find close to optimal solutions. In this thesis, we study algorithms for computing such approximate solutions