We study the computational complexity of candidate control in elections with few voters, that is, we consider the parameterized complexity of candidate control in elections with respect to the number of voters as a parameter. We consider both the standard scenario of adding and deleting candidates, where one asks whether a given candidate can become a winner (or, in the destructive case, can be precluded from winning) by adding or deleting few candidates, as well as a combinatorial scenario where adding/deleting a candidate automatically means adding or deleting a whole group of candidates. Considering several fundamental voting rules, our results show that the parameterized complexity of candidate control, with the number of voters as the parameter, is much more varied than in the setting with many voters.

Election control problems are concerned with affecting the result of an election by modifying the structure of the election. Such election modifications could be either introducing some new candidates or voters or removing some existing candidates or voters from the election or partitioning candidates or voters [2, 27, 32, 42, 56, 57, 34, 35, 62]. We focus on the computational complexity of election control by adding and deleting candidates (that is, candidate control), for the case where the election involves only a few voters. From the viewpoint of computational complexity, voter control with few voters has not received sufficient study. We focus on very simple, practical voting rules such as Plurality, Veto, andt-Approval, but discuss several more involved rules as well. To analyze the effect of allowing only a small number of voters, we use the formal tools of parameterized complexity theory [21, 23, 38, 60]. From the viewpoint of classical complexity theory, most of the candidate control problems for most of the typically studied voting rules are NPhard. Indeed, candidate control problems are NPhard even for the Plurality rule; nonetheless, there are some natural examples of candidate control problems with polynomialtime algorithms. It turns out that for the case of elections with few voters, that is, for control problems parameterized by the number of voters, the computational complexity landscape of candidate control is much more varied and sometimes quite surprising.

Control and bribery are settings in which an external agent seeks to influence the outcome of an election. Constructive control of elections refers to attempts by an agent to, via such actions as addition/deletion/partition of candidates or voters, ensure that a given candidate wins. Destructive control refers to attempts by an agent to, via the same actions, preclude a given candidate's victory. An election system in which an agent can sometimes affect the result and it can be determined in polynomial time on which inputs the agent can succeed is said to be vulnerable to the given type of control. An election system in which an agent can sometimes affect the result, yet in which it is NP-hard to recognize the inputs on which the agent can succeed, is said to be resistant to the given type of control. Aside from election systems with an NP-hard winner problem, the only systems previously known to be resistant to all the standard control types were highly artificial election systems created by hybridization. This paper studies a parameterized version of Copeland voting, denoted by Copeland^\alpha, where the parameter \alpha is a rational number between 0 and 1 that specifies how ties are valued in the pairwise comparisons of candidates. In every previously studied constructive or destructive control scenario, we determine which of resistance or vulnerability holds for Copeland^\alpha for each rational \alpha, 0 \leq \alpha \leq 1. In particular, we prove that Copeland^{0.5}, the system commonly referred to as ``Copeland voting,'' provides full resistance to constructive control, and we prove the same for Copeland^\alpha, for all rational \alpha, 0 < \alpha < 1. Among systems with a polynomial-time winner problem, Copeland voting is the first natural election system proven to have full resistance to constructive control. In addition, we prove that both Copeland^0 and Copeland^1 (interestingly, Copeland^1 is an election system developed by the thirteenth-century mystic Llull) are resistant to all standard types of constructive control other than one variant of addition of candidates. Moreover, we show that for each rational \alpha, 0 \leq \alpha \leq 1, Copeland^\alpha voting is fully resistant to bribery attacks, and we establish fixed-parameter tractability of bounded-case control for Copeland^\alpha. We also study Copeland^\alpha elections under more flexible models such as microbribery and extended control, we integrate the potential irrationality of voter preferences into many of our results, and we prove our results in both the unique-winner model and the nonunique-winner model. Our vulnerability results for microbribery are proven via a novel technique involving min-cost network flow.