Voting and bribing in single-exponential time

Abstract

We introduce a general problem about bribery in voting systems. In theR-Multi-Briberyproblem,the goal is to bribe a set of voters at minimum cost such that a desired candidate wins the perturbedelection under the voting ruleR. Voters assign prices for withdrawing their vote, for swapping thepositions of two consecutive candidates in their preference order, and for perturbing their approval countto favour candidates.As our main result, we show thatR-Multi-Briberyis fixed-parameter tractable parameterized bythe number of candidates for many natural voting rulesR, including Kemeny rule, all scoring protocols,maximin rule, Bucklin rule, fallback rule, SP-AV, and any C1 rule. In particular, our result resolves theparameterized complexity ofR-Swap Briberyfor all those voting rules, thereby solving a long-standingopen problem and “Challenge #2” of the “Nine Research Challenges in Computational Social Choice”by Bredereck et al.Further, our algorithm runs in single-exponential time for arbitrary cost; it thus improves the earlierdouble-exponential time algorithm by Dorn and Schlotter that is restricted to the uniform-cost case forall scoring protocols, the maximin rule, and Bucklin rule.