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

Abstract

We study the computational complexity of committee selection problem for several approval-based voting rules in the presence of outliers. Our first result shows that outlier consideration makes committee selection problem intractable for approval, net approval, and minisum approval voting rules. We then study parameterized complexity of this problem with five natural parameters, namely the target score, the size of the committee (and its dual parameter, the number of candidates outside the committee), the number of outliers (and its dual parameter, the number of non-outliers). For net approval and minisum approval voting rules, we provide a dichotomous result, resolving the parameterized complexity of this problem for all subsets of five natural parameters considered (by showing either FPT or W[1]-hardness for all subsets of parameters). For the approval voting rule, we resolve the parameterized complexity of this problem for all subsets of parameters except one.We also study approximation algorithms for this problem. We show that there does not exist any α(·) factor approximation algorithm for approval and net approval voting rules, for any computable function α(·), unless P = NP. For the minisum voting rule, we provide a (1 + ε) factor approximation algorithm running in time n O(log m/ε 2 ) , for every constant ε > 0, where m and n are the number of candidates and the number of votes respectively.

Computational Social Choice and Computational Complexity: BFFs?

“…We have discussed just a few of the rich collection of interactions between comsoc and computational complexity. Due to space limitations, many others-from how the complexity-theoretic join operation allows proofs in comsoc of the impossibility of proving certain impossibility theorems (Hemaspaandra, Hemaspaandra, and Rothe 2009) to how work on online control gives an unexpected new quantifier-alternation characterization of the class coNP (Hemaspaandra, Hemaspaandra, and Rothe 2017) to the rich interactions with approximation (see as just one of many examples Faliszewski, Skowron, and Talmon, 2017) to the power of dichotomy results to the insights given by parameterized complexity (both these last are on view simultaneously, for example, in Dey, Misra, and Narahari, 2017) to much more-are not even touched on here.…”

Computational Social Choice and Computational Complexity: BFFs?

“…We have discussed just a few of the rich collection of interactions between comsoc and computational complexity. Many others-from how the complexity-theoretic join operation allows proofs in comsoc of the impossibility of proving certain impossibility theorems [HHR09] to how work on online control gives an unexpected new quantifier-alternation characterization of the class coNP [HHR17] to the rich interactions with approximation (see as just one of many examples [FST17]) to the power of dichotomy results to the insights given by parameterized complexity (both these last are on view simultaneously, for example, in [DMN17]) to much more-are not even touched on here.…”

Gerrymandering on graphs: Computational complexity and parameterized algorithms

“…Due to space constraints we do not discuss them here. Parameterized complexity of manipulation has received extensive attention over the last several years, [1,2,16,17,10] are just a few examples.…”