Complexity of Optimal Lobbying in Threshold Aggregation

Optimal Lobbying is the problem a lobbyist or a campaign manager faces in a full-information voting scenario of a multiissue 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 must influence to change their vote, for a desired outcome to be obtained. We study the computational complexity of Optimal Lobbying when the issues are aggregated using an anonymous monotone function and the family of desired outcomes is an upward-closed family. 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. We show that for extreme values of the parameters, the problem is tractable, and provide algorithms. On the other hand, we prove intractability of the problem for the complementary cases, which are most of the values of the parameters.