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Possible winner problems on partial tournaments: a parameterized study

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Abstract

We study possible winner problems related to the uncovered set and the Banks set on partial tournaments from the viewpoint of parameterized complexity. We first study a 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. We focus on two parameterizations: parameterized by |X| and parameterized by the number of arcs to be added. In addition, we study a parameterized variant of the problem which is to determine whether all vertices of X can 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 the Banks set. We achieve \(\mathcal {XP}\) results, \(\mathcal {W}\)-hardness results as well as \(\mathcal {FPT}\) results along with a kernelization lower bound for the problems studied in this paper.

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Notes

  1. In Aziz et al. (2015), the authors use \(\text {PSW}_\text {UC}\) to denote the problem.

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Acknowledgments

We sincerely thank the anonymous reviewers of ADT 2013 for their constructive comments.

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Authors and Affiliations

  1. Universität des Saarlandes, Saarbrücken, Germany

    Yongjie Yang

  2. School of Computer Science and Technology, Shandong University, Jinan, China

    Jiong Guo

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  1. Yongjie Yang
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  2. Jiong Guo
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Correspondence to Yongjie Yang.

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Yang, Y., Guo, J. Possible winner problems on partial tournaments: a parameterized study. J Comb Optim 33, 882–896 (2017). https://doi.org/10.1007/s10878-016-0012-1

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