We study the complexity of Destructive Shift Bribery. In this problem, we are given an election with a set of candidates and a set of voters (each ranking the candidates from the best to the worst), a despised candidate @math , a budget @math , and prices for shifting @math back in the voters' rankings. The goal is to ensure that @math is not a winner of the election. We show that this problem is polynomial-time solvable for scoring protocols (encoded in unary), the Bucklin and Simplified Bucklin rules, and the Maximin rule, but is NP-hard for the Copeland rule. This stands in contrast to the results for the constructive setting (known from the literature), for which the problem is polynomial-time solvable for @math -Approval family of rules, but is NP-hard for the Borda, Copeland, and Maximin rules. We complement the analysis of the Copeland rule showing W-hardness for the parameterization by the budget value, and by the number of affected voters. We prove that the problem is W-hard when parameterized by the number of voters even for unit prices. From the positive perspective we provide an efficient algorithm for solving the problem parameterized by the combined parameter the number of candidates and the maximum bribery price (alternatively the number of different bribery prices).
A variant of was studied by @cite_38 in their work on testing how robust are winners of given elections. They presented an analysis of the case where every swap has a unit price and they showed that the problem is easy for scoring protocols and for the Condorcet rule. The authors' definition of the Condorcet rule slightly differs from a standard one usually seen in the literature. They assume that if there is no unique Condorcet winner the rule returns the whole set of candidates instead of an empty set (well established as a return value for this case in the literature.) This work is very closely related to ours, but the definition of the problem is somewhat different (more general types of swaps but less general price functions). Very recently @cite_19 studied an analogous setting ( with unit prices) for the case of multiwinner voting rules.
@cite_19 @cite_38
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We investigate how robust the results of committee elections are to small changes in the input preference orders, depending on the voting rules used. We find that for typical rules the effect of making a single swap of adjacent candidates in a single preference order is either that (1) at most one committee member might be replaced, or (2) it is possible that the whole committee will be replaced. We also show that the problem of computing the smallest number of swaps that lead to changing the election outcome is typically NP-hard, but there are natural FPT algorithms. Finally, for a number of rules we assess experimentally the average number of random swaps necessary to change the election result. We study the sensitivity of election outcomes to small changes in voters' preferences. We assume that a voter may err by swapping two adjacent candidates in his vote; we would like to check whether the election outcome would remain the same given up to delta errors. We show that this problem can be viewed as the destructive version of the unit-cost swap bribery problem, and demonstrate that it is polynomial-time solvable for all scoring rules as well as for the Condorcet rule. We are also interested in identifying elections that are maximally robust with respect to a given voting rule. We define the robustness radius of an election with respect to a given voting rule as the maximum number of errors that can be made without changing the election outcome; the robustness of a voting rule is defined as the robustness radius of the election that is maximally robust with respect to this rule. We derive bounds on the robustness of various voting rules, including Plurality, Borda, and Condorcet.