Manipulation with randomized tie-breaking under Maximin

In recent papers, Obraztsova et al. initiated the study of the computational complexity of voting manipulation under randomized tie-breaking [3, 2]. The authors provided a polynomial-time algorithm for the problem of finding an optimal vote for the manipulator (a vote maximizing the manipulator's expected utility) under the Maximin voting rule, for the case where the manipulator's utilities of the candidates are given by the vector (1, 0,...,0). On the other hand, they showed that this problem is NP-hard for the case where the utilities are (1,...,1, 0).
This paper continues that line of research. We prove that when the manipulator's utilities of the candidates are given by the vector (1,...,1, 0,...,0), with k 1's and (m -- k) 0's, then the problem of finding an optimal vote for the manipulator is fixed-parameter tractable when parameterized by k. Also, by exploring the properties of the graph built by the algorithm, we prove that when a certain sub-graph of this graph contains a 2-cycle, then the solution returned by the algorithm is optimal.