Embedding Phylogenetic Trees in Networks of Low Treewidth

Given a rooted, binary phylogenetic network and a rooted, binary phylogenetic tree, can the tree be embedded into the network? This problem, called Tree Containment, arises when validating networks constructed by phylogenetic inference methods. We present the first algorithm for (rooted) Tree Containment using the treewidth t of the input network N as parameter, showing that the problem can be solved in 2O(t2) |N| time and space.Optimizatio

Elements of dynamic and 2-SAT programming: paths, trees, and cuts

This thesis presents faster (in terms of worst-case running times) exact algorithms for special cases of graph problems through dynamic programming and 2-SAT programming. Dynamic programming describes the procedure of breaking down a problem recursively into overlapping subproblems, that is, subproblems with common subsubproblems. Given optimal solutions to these subproblems, the dynamic program then combines them into an optimal solution for the original problem. 2-SAT programming refers to the procedure of reducing a problem to a set of 2-SAT formulas, that is, boolean formulas in conjunctive normal form in which each clause contains at most two literals. Computing whether such a formula is satisfiable (and computing a satisfying truth assignment, if one exists) takes linear time in the formula length. Hence, when satisfying truth assignments to some 2-SAT formulas correspond to a solution of the original problem and all formulas can be computed efficiently, that is, in polynomial time in the input size of the original problem, then the original problem can be solved in polynomial time. We next describe our main results. Diameter asks for the maximal distance between any two vertices in a given undirected graph. It is arguably among the most fundamental graph parameters. We provide both positive and negative parameterized results for distance-from-triviality-type parameters and parameter combinations that were observed to be small in real-world applications. In Length-Bounded Cut, we search for a bounded-size set of edges that intersects all paths between two given vertices of at most some given length. We confirm a conjecture from the literature by providing a polynomial-time algorithm for proper interval graphs which is based on dynamic programming. k-Disjoint Shortest Paths is the problem of finding (vertex-)disjoint paths between given vertex terminals such that each of these paths is a shortest path between the respective terminals. Its complexity for constant k > 2 has been an open problem for over 20 years. Using dynamic programming, we show that k-Disjoint Shortest Paths can be solved in polynomial time for each constant k. The problem Tree Containment asks whether a phylogenetic tree T is contained in a phylogenetic network N. A phylogenetic network (or tree) is a leaf-labeled single-source directed acyclic graph (or tree) in which each vertex has in-degree at most one or out-degree at most one. The problem stems from computational biology in the context of the tree of life (the history of speciation). We introduce a particular variant that resembles certain types of uncertainty in the input. We show that if each leaf label occurs at most twice in a phylogenetic tree N, then the problem can be solved in polynomial time and if labels can occur up to three times, then the problem becomes NP-hard. Lastly, Reachable Object is the problem of deciding whether there is a sequence of rational trades of objects among agents such that a given agent can obtain a certain object. A rational trade is a swap of objects between two agents where both agents profit from the swap, that is, they receive objects they prefer over the objects they trade away. This problem can be seen as a natural generalization of the well-known and well-studied Housing Market problem where the agents are arranged in a graph and only neighboring agents can trade objects. We prove a dichotomy result that states that the problem is polynomial-time solvable if each agent prefers at most two objects over its initially held object and it is NP-hard if each agent prefers at most three objects over its initially held object. We also provide a polynomial-time 2-SAT program for the case where the graph of agents is a cycle

Focus: A Graph Approach for Data-Mining and Domain-Specific Assembly of Next Generation Sequencing Data

Next Generation Sequencing (NGS) has emerged as a key technology leading to revolutionary breakthroughs in numerous biomedical research areas. These technologies produce millions to billions of short DNA reads that represent a small fraction of the original target DNA sequence. These short reads contain little information individually but are produced at a high coverage of the original sequence such that many reads overlap. Overlap relationships allow for the reads to be linearly ordered and merged by computational programs called assemblers into long stretches of contiguous sequence called contigs that can be used for research applications. Although the assembly of the reads produced by NGS remains a difficult task, it is the process of extracting useful knowledge from these relatively short sequences that has become one of the most exciting and challenging problems in Bioinformatics. The assembly of short reads is an aggregative process where critical information is lost as reads are merged into contigs. In addition, the assembly process is treated as a black box, with generic assembler tools that do not adapt to input data set characteristics. Finally, as NGS data throughput continues to increase, there is an increasing need for smart parallel assembler implementations. In this dissertation, a new assembly approach called Focus is proposed. Unlike previous assemblers, Focus relies on a novel hybrid graph constructed from multiple graphs at different levels of granularity to represent the assembly problem, facilitating information capture and dynamic adjustment to input data set characteristics. This work is composed of four specific aims: 1) The implementation of a robust assembly and analysis tool built on the hybrid graph platform 2) The development and application of graph mining to extract biologically relevant features in NGS data sets 3) The integration of domain specific knowledge to improve the assembly and analysis process. 4) The construction of smart parallel computing approaches, including the application of energy-aware computing for NGS assembly and knowledge integration to improve algorithm performance. In conclusion, this dissertation presents a complete parallel assembler called Focus that is capable of extracting biologically relevant features directly from its hybrid assembly graph

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum