| In this thesis we study the feedback set problems in directed graphs. The feedback set problems deal with destroying (directed) cycles in (directed) graphs using a minimum number of vertex or edge removals. These problems originated from applications in combinatorial circuit design, but have found their way into numerous other applications such as deadlock prevention in operating systems, constraint satisfaction and bayesian inference in artificial intelligence and graph drawing. Therefore, in recent years feedback set problems have been the subject of growing interest and there have been intensive efforts on exact and approximation algorithms for these kinds of problems.; This thesis focuses on feedback set problems in a special class of directed graphs, tournament and bipartite tournament, which appear in applications such as voting systems, rankings, and graph drawing. We briefly survey the best known hardness results, parameterized complexity results, and approximation algorithms for feedback set problems in a tournament. For a bipartite tournament, we present an improved 3-approximation algorithm and an improved O(3kn2) time FPT algorithm for the feedback vertex set problem in a bipartite tournament. We also present some new results concerning the computational complexity of the feedback arc set problem in a bipartite tournament and provide an improved FPT algorithm for the feedback arc set problem in a bipartite tournament.; Keywords: Feedback Set Problem, Tournament, Parameterized Complexity, FPT algorithm, kernelization, Linear programming, Approximation Algorithm, Iterative Rounding. |
