We developed a unified SAT-solving approach to three phylogenetic network construction problems that integer linear programming was not able to solve. We detail the CNF formulations for two: the Minimum-Reticulation Problem, and the Hybridization-Network Problem; and apply them to widely studied Grass data, and to randomly generated data. The results reduce the number of reticulations in 4 of the 18 Grass datasets, and improve the best SAT-based solutions in two more. We study the effectiveness of a method for speeding the solution to the Hybridization problem, and show that when it can be applied, it can be extremely effective. This work contributes to SAT-solving methodology, to phylogenetic network construction, and to the evolutionary history of Grasses.

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The goal of the tutorial is to introduce and detail *modeling* and *solving* of real problems in computational and systems biology using integer linear programming. This is in contrast to goal of explaining integer programming theory, or how integer programming solvers work.

The goal of the tutorial is to introduce and detail *modeling* and *solving* of real problems in computational and systems biology using integer linear programming. This is in contrast to goal of explaining integer programming theory, or how integer programming solvers work.

Tanglegrams are a tool to infer joint evolution of species. Tanglegrams are widely used in ecology to study joint evolution history of parasitic or symbiotically linked species. Visually, a tanglegram is a pair of evolutionary trees drawn with the leaves facing at each other. One species at the leaf of one trees is related ecologically to a species at a leaf of another tree. Related species from the two trees are connected by an edge. The number of crossings between the edges joining the leaves indicate the relatedness of the trees. Earlier work on tanglegrams considered the same number of leaves on both the trees and one edge between the leaves of the two trees. In this paper we consider multiple edges from a leaf in the trees. These edges correspond to ecological events like duplication, host switching etc. We generalize the definition of tanglegrams to admit multiple edges between the leaves. We show integer programs for optimizing the number of crossings. The integer program has an XOR formulation very similar to the formulation for the tanglegrams. We also show how the ideas for distance minimization on tanglegrams can be extended for the generalized tanglegrams. We show that the tanglegram drawings used in ecology can be improved to have fewer crossings using our integer programs.

Evolutionary data has been traditionally modeled via phylogenetic trees; however, branching alone cannot model conflicting phylogenetic signals, so networks are used instead. Ancestral recombination graphs (ARGs) are used to model the evolution of incompatible sets of SNP data, allowing each site to mutate only once. The model often aims to minimize the number of recombinations. Similarly, incompatible cluster data can be represented by a reticulation network that minimizes reticulation events. The ARG literature has traditionally been disjoint from the reticulation network literature. By building on results from the reticulation network literature, we resolve an open question of interest to the ARG community. We explicitly prove that the History Bound, a lower bound on the number of recombinations in an ARG for a binary matrix, which was previously only defined procedurally, is equal to the minimum number of reticulation nodes in a network for the corresponding cluster data. To facilitate the proof, we give an algorithm that constructs this network using intermediate values from the procedural History Bound definition. We then develop a top-down algorithm for computing the History Bound, which has the same worst-case runtime as the known dynamic program, and show that it is likely to run faster in typical cases.

The basic RNA Secondary Structure Prediction problem or Single Sequence Folding Problem (SSF) was solved thirty-five years ago by a now well-known O(n^3) -time dynamic programming method. Recently three methodologies - Valiant, Four-Russians, and Sparsification - have been applied to speedup RNA Secondary Structure prediction. In this paper we combine the previously independent speedups of Sparsification and Four-Russians. The Sparsification method exploits two properties of the input: the number of subsequence Z with the endpoints belonging to the optimal folding set and the maximum number base-pairs L. These sparsity properties satisfy 0 ≤ L ≤ n / 2 and n ≤ Z ≤ n^2 / 2 , and the method reduces the algorithmic running time to O(LZ). In this paper, we first reformulate the SSF Four-Russians (n^3/log ^2 n) -time algorithm, implied by Pinhas et al. [24], to utilize an on-demand lookup table. This formulation not only removes all extraneous computation and allows us to incorporate more realistic scoring schemes, but leads us to take advantage of the sparsity properties. Our main result is a framework that combines the fastest Sparsification and fastest Four-Russians Methods. For SSF, this combined method has worst-case running time of O(L̃Z̃) , where L/log n≤L̃≤ min(L,n/log n) and Z/log n≤Z̃≤ min(Z,n^2/log n) . Through asymptotic analysis and empirical testing on the base-pair maximization variant, we show that this framework is able to achieve a speedup on every problem instance, that is asymptotically never worse, and empirically better than achieved by the minimum of the two methods alone.

For many important complex traits, Genome Wide Association Studies (GWAS) have only recovered a small proportion of the variance in disease prevalence known to be caused by genetics. The most common explanation for this is the presence of multiple rare mutations that cannot be identified in GWAS due to a lack of statistical power. Such rare mutations may be concentrated in relatively few genes, as is the case for many known Mendelian diseases, where the mutations are often compound heterozygous (CH), defined below. Due to the multiple mutations, each of which contributes little by itself to the prevalence of the disease, GWAS also lacks power to identify genes contributing to a CH-trait. In this paper, we address the problem of finding genes that are causal for CH-traits, by introducing a discrete optimization problem, called the Phenotypic Distance Problem. We show that it can be efficiently solved on realistic-size simulated CH-data by using integer linear programming (ILP). The empirical results strongly validate this approach.