Abstract. In voting theory, bribery is a form of manipulative behavior in which an external actor (the briber) offers to pay the voters to change their votes in order to get her preferred candidate elected. We investigate a model of bribery where the price of each vote depends on the amount of change that the voter is asked to implement. Specifically, in our model the briber can change a voter’s preference list by paying for a sequence of swaps of consecutive candidates. Each swap may have a different price; the price of a bribery is the sum of the prices of all swaps that it involves. We prove complexity results for this model, which we call swap bribery, for a broad class of voting rules, including variants of approval and k-approval, Borda, Copeland, and maximin. 1

We study the computational complexity of the counting version of the POSSIBLE-WINNER problem for elections. In the POSSIBLE-WINNER problem we are given a profile of voters, each with a partial preference order, and ask if there are linear extensions of the votes such that a designated candidate wins. We also analyze a special case of POSSIBLE-WINNER, the MANIPULATION problem. We provide polynomial-time algorithms for counting manipulations in a class of scoring protocols and in several other voting rules. We show #Phardness of the counting variant of POSSIBLE-WINNER for plurality and veto and give a simple yet general and practically useful randomized algorithm for a variant of POSSIBLE-WINNER for all voting rules for which a winner can be computed in polynomial time.

In a voting system, sometimes multiple new alternatives will join the election after the voters’ preferences over the initial alternatives have been revealed. Computing whether a given alternative can be a co-winner when multiple new alternatives join the election is called the possible co-winner with new alternatives (PcWNA) problem, introduced by Chevaleyre et al. [4, 5]. In this paper, we show that the PcWNA problems are NP-complete for the Bucklin, Copeland0, and Simpson (a.k.a. maximin) rule, even when the number of new alternatives is no more than a constant. We also show that the PcWNA problem can be solved in polynomial time for plurality with runoff. For the approval rule, we define three different ways to extend a linear order with new alternatives, and characterize the computational complexity of the PcWNA problem for each of them. 1

Research on parameterized algorithmics for NP-hard problems has steadily grown over the last years. We survey and discuss how parameterized complexity analysis naturally develops into the field of multivariate algorithmics. Correspondingly, we describe how to perform a systematic investigation and exploitation of the “parameter space” of computationally hard problems.

Abstract. Research on parameterized algorithmics for NP-hard problems has steadily grown over the last years. We survey and discuss how parameterized complexity analysis naturally develops into the field of multivariate algorithmics. Correspondingly, we describe how to perform a systematic investigation and exploitation of the “parameter space ” of computationally hard problems.

Winner determination in voting trees with incomplete