Parameterized dichotomy of choosing committees based on approval votes in the presence of outliers

Approval voting provides an opportunity for the agents to make a comment about every candidate, without incurring the overhead of determining a full ranking on the entire set of candidates. This makes approval voting a natural choice for many practical applications. In this work, we focus on the use of approval voting for selecting a committee in scenarios where we can have few outrageous voters whom we call outliers. More specifically, we study the computational complexity of the committee selection problem for commonly used approval-based voting rules in the presence of outliers. Our first result shows that outliers render the committee selection problem intractable for approval, net approval, and minisum approval voting rules. We next study the parameterized complexity of this problem with five natural parameters, namely the target score, the size of the committee (and its dual parameter namely the number of candidates outside the committee); and the number of outliers (and its dual parameter namely the number of non-outliers). Our main contribution in this paper is dichotomous results, which establish the parameterized complexity of the problem of selecting a committee under approval, net approval, and minisum approval voting rules for all subsets of the above five parameters considered here (by showing either FPT or W[1] -hardness for all subsets of parameters).