We study possible winner problems related to uncovered set and Banks set on partial tournaments from the viewpoint of parameterized complexity. We first study the following problem, where given a partial tournament D and a subset X of vertices, we are asked to add some arcs to D such that all vertices in X are included in the uncovered set. Here we focus on two parameterizations of the problem: parameterized by |X| and parameterized by the number of arcs to be added to make all vertices of X be included in the uncovered set. In addition, we study a parameterized variant of the problem to decide whether we can make all vertices of X be included in the uncovered set by reversing at most k arcs. Finally, we study some parameterizations of a possible winner problem on partial tournaments, where we are given a partial tournament D and a distinguished vertex p, and asked whether D has a maximal transitive subtournament with p being the 0-indegree vertex. These parameterized problems are related to Banks set. For all these parameterized problems studied in this paper, we achieve XP results, W-hardness results as well as FPT results along with a kernelization lower bound.
展开▼
机译:从参数化复杂性的角度来看,我们研究了与部分锦标赛上的未发现集和银行相关的赢家问题。我们首先研究下列问题,在给定部分锦标赛D和顶点子集X的情况下,我们被要求将一些弧添加到D,使得X中的所有顶点都包含在未覆盖的集合中。在这里,我们专注于问题的两个参数化:参数化由| x |并由要添加的弧数进行参数化以使X的所有顶点都包含在未覆盖的集合中。此外,我们研究了问题的参数化变体,以决定我们是否可以通过最多k弧线颠倒在未覆盖的设置中将X的所有顶点都包含在未覆盖的设置中。最后,我们研究了部分锦标赛中可能的胜利问题的一些参数化,在那里我们被赋予了部分锦标赛D和一个杰出的顶点P,并询问D是否具有P的最大传递子类别,具有p为0-Indegree顶点。这些参数化问题与银行集合有关。对于本文研究的所有这些参数化问题,我们实现了XP结果,硬度结果以及FPT结果以及内孔的下限。
展开▼