Complexity of optimal lobbying in threshold aggregation

This work studies the computational complexity of Optimal Lobbying under Threshold Aggregation. Optimal Lobbying is the problem a lobbyist or a campaign manager faces in a voting scenario of a multi-issue referendum when trying to influence the result. The Lobby is faced with a profile that specifies for each voter and each issue whether the voter approves or rejects the issue, and seeks to find the smallest set of voters it can influence to change their vote, for a desired outcome to be obtained. This problem also describes problems arising in other scenarios of aggregation, such as principal-agents incentives scheme in a complex combinatorial problem, and bribery in Truth-Functional Judgement Aggregation. We study cases when the issues are aggregated by a threshold aggregator, that is, an anonymous monotone function, and the desired outcomes set is upward-closed. We analyze this problem with regard to two parameters: the minimal number of supporters needed to pass an issue, and the size of the maximal minterm of the desired set. For these parameters we separate tractable cases from untractable cases and in that generalize the NP-complete result of Christian et al. [8]. We show that for the extreme values of the parameters, the problem is solvable in polynomial time, and provide algorithms. On the other hand, we prove the problem is not solvable in polynomial time for the non-extremal values, which are common values for the parameters.