Piotr Skowron

Properties of multiwinner voting rules

A committee selection rule (or, multiwinner voting rule) is a mapping that takes a collection of strict preference rankings and a positive integer k as input, and outputs one or more subsets of candidates of size k. In this paper we consider committee selection rules that can be viewed as generalizations of single-winner scoring rules, including SNTV, Bloc, k-Borda, STV, as well as several variants of the Chamberlin–Courant rule and the Monroe rule and their approximations. We identify two natural broad classes of committee selection rules, and show that many of the existing rules belong to one or both of these classes. We then formulate a number of desirable properties of committee selection rules, and evaluate the rules we consider with respect to these properties.

The complexity of fully proportional representation for single-crossing electorates

We study the complexity of winner determination in single-crossing elections under two classic fully proportional representation rules-Chamberlin-Courants rule and Monroes rule. Winner determination for these rules is known to be NP-hard for unrestricted preferences. We show that for single-crossing preferences this problem admits a polynomial-time algorithm for Chamberlin-Courants rule, but remains NP-hard for Monroes rule. Our algorithm for Chamberlin-Courants rule can be modified to work for elections with bounded single-crossing width. We then consider elections that are both single-peaked and single-crossing, and develop an efficient algorithm for the egalitarian variant of Monroes rule for such elections. While Betzler et al. [3] have recently presented a polynomial-time algorithm for this rule under single-peaked preferences, our algorithm has considerably better worst-case running time than that of Betzler et al.

The Complexity of Fully Proportional Representation for Single-Crossing Electorates

We study the complexity of winner determination in single-crossing elections under two classic fully proportional representation rules—Chamberlin–Courant’s rule and Monroe’s rule. Winner determination for these rules is known to be NP-hard for unrestricted preferences. We show that for single-crossing preferences this problem admits a polynomial-time algorithm for Chamberlin–Courant’s rule, but remains NP-hard for Monroe’s rule. Our algorithm for Chamberlin–Courant’s rule can be modified to work for elections with bounded single-crossing width. To circumvent the hardness result for Monroe’s rule, we consider single-crossing elections that satisfy an additional constraint, namely, ones where each candidate is ranked first by at least one voter (such elections are called narcissistic). For single-crossing narcissistic elections, we provide an efficient algorithm for the egalitarian version of Monroe’s rule.

Elections with Few Candidates: Prices, Weights, and Covering Problems

We show that a number of election-related problems with prices such as, for example, bribery are fixed-parameter tractable in

The Condorcet Principle for Multiwinner Elections: From Shortlisting to Proportionality

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Equilibria of Plurality Voting: Lazy and Truth-Biased Voters

when parameterized by the number of candidates. For bribery, this resolves a nearly 10-year old family of open problems. Our results follow by a general technique that formulates voting problems as covering problems and extends the classic approach of using integer linear programming and the algorithm of Lenstrai¾?[19]. In this context, our central result is that Weighted Set Multicover parameterized by the universe size is fixed-parameter tractable. Our approach is also applicable to weighted electoral control for Approval voting. We improve previously known

FPT approximation schemes for maximizing submodular functions

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