Fixed-parameter algorithms for some combinatorial problems in bioinformatics

NP-hard problems occur often in bioinformatics. The typical huge sizes of biological problem instances often prohibit solving NP-hard problems in bioinformatics optimally with classical approaches. Fixed-parameterized algorithmics has been developed in 1990s as a new approach to solve NP-hard problem optimally in a guaranteed running time. The essential idea of fixed-parameter algorithms is to restrict the combinatorial explosion of the solution space of an NP-hard problem to a small input parameter, instead of the size of the problem instance. Thus, fixed-parameter algorithms offer a new opportunity to solve NP-hard problems with large instances. In this thesis, we apply fixed-parameter algorithms to cope with three NP-hard problems in bioinformatics: • Flip Consensus Tree Problem is a special case of the Flip Supertree Problem, a combinatorial problem arising in computational phylogenetics. Using a graph-theoretical formulation of the Flip Consensus Tree Problem that asks for a minimum set of edge modifications to transform a bipartite graph into an M-free bipartite graph, we present a set of data reduction rules and two depthbounded search tree algorithms for this problem that are fixed-parameter with respect to the minimum number of edge modifications. Additionally, we discuss several heuristic improvements to accelerate the running time of our algorithms in practice. We also report computational results on phylogenetic data. • Weighted Cluster Editing Problem is a graph-theoretical problem, that asks for a set of edge modifications with minimum cost to transform a graph into a cluster graph. This problem often arises in computational biology when clustering objects with respect to a given similarity or distance measure is required. We present one of our depth-bounded search tree algorithms for this problem that is a fixed-parameter algorithm with respect to the minimum modification cost. We also describe the main idea of our fastest algorithm [16, 18] for the Weighted Cluster Editing Problemand Unweighted Cluster Editing Problem. • Bond Order Assignment Problem asks for a bond order assignment of a molecule graph that minimizes a penalty function. We show that the Bond Order Assignment Problem is NP-hard and inapproximable unless P=NP. Furthermore, we show that the maximization version of Bond Order Assignment Problem is MAX SNP-hard. We then give two exact fixed-parameter algorithms for the problem, where bond orders are computed via dynamic programming on a tree decomposition of the molecule graph. Our algorithms are fixed-parameter