Very Hard Electoral Control Problems

It is important to understand how the outcome of an election can be modified by an agent with control over the structure of the election. Electoral control has been studied for many election systems, but for all studied systems the winner problem is in P, and so control is in NP. There are election systems, such as Kemeny, that have many desirable properties, but whose winner problems are not in NP. Thus for such systems control is not in NP, and in fact we show that it is typically complete for $\Sigma_2^p$ (i.e., ${\rm NP}^{\rm NP}$, the second level of the polynomial hierarchy). This is a very high level of complexity. Approaches that perform quite well for solving NP problems do not necessarily work for $\Sigma_2^p$-complete problems. However, answer set programming is suited to express problems in $\Sigma_2^p$, and we present an encoding for Kemeny control.