In the Shift Bribery problem, we are given an election (based on preference orders), a preferred candidate p, and a budget. The goal is to ensure that p wins by shifting p higher in some voters' preference orders. However, each such shift request comes at a price (depending on the voter and on the extent of the shift) and we must not exceed the given budget. We study the parameterized computational complexity of Shift Bribery with respect to a number of parameters (pertaining to the nature of the solution sought and the size of the election) and several classes of price functions. When we parameterize Shift Bribery by the number of affected voters, then for each of our voting rules (Borda, Maximin, Copeland) the problem is W[2]-hard. If, instead, we parameterize by the number of positions by which p is shifted in total, then the problem is fixed-parameter tractable for Borda and Maximin, and is W[1]-hard for Copeland. If we parameterize by the budget, then the results depend on the price function class. We also show that Shift Bribery tends to be tractable when parameterized by the number of voters, but that the results for the number of candidates are more enigmatic. 1 For example, the German website idealo.de aggregates different product tests by first translating the test results into a unified rating system and then taking the "average" of all the ratings. Various university rankings are prepared in a similar way. It would be very interesting, however, to utilize the rankings themselves, instead of the ratings, for the aggregation. Moreover, Formula 1 racing (and numerous similar competitions) use pure ranking information (e.g., Formula 1 uses a very slightly modified variant of the Borda election rule). 2 We assume that we have the knowledge of the voters' preference orders (for example, from preelection polls). Further, in our example settings often the full rankings are known. For example, a driver preparing for a new Formula 1 season has full knowledge of the results from the previous one. 3 What "to convince" means can vary a lot depending on the application scenario. On the evil side we have bribery, but it can also mean things such as product development, hiring more faculty members, training on a particular racing circuit, or explaining the details of one's political platform. Clearly, different ranking providers may appreciate different efforts, which is modeled by the individual price functions.
Chen et al. REF considered the parametrized complexity of Constructive Shift Bribery, and have shown a varied set of results (in general, parametrization by the number of positions by which the preferred candidate is shifted tends to lead to FPT-time algorithms, parametrization by the number of affected voters tends to lead to hardness results, and parametrization by the available budget gives results between these two extremes).
Prices Matter for the Parameterized Complexity of Shift Bribery
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Computer Science Mathematics