A parameterized perspective on protecting elections

We study the parameterized complexity of the OPTIMAL DEFENSE and OPTIMAL ATTACK problems in voting. In both the problems, the input is a set of voter groups (every voter group is a district consisting of a set of votes) and two integers k<inf>a</inf> and k<inf>d</inf> corresponding to respectively the number of voter groups the attacker can attack and the number of voter groups the defender can defend. A voter group gets removed from the election if it is attacked but not defended. In the OPTIMAL DEFENSE problem, we want to know if it is possible for the defender to commit to a strategy of defending at most k<inf>d</inf> voter groups such that, no matter which k<inf>a</inf> voter groups the attacker attacks, the outcome of the election does not change. In the OPTIMAL ATTACK problem, we want to know if it is possible for the attacker to commit to a strategy of attacking k<inf>a</inf> voter groups such that, no matter which k<inf>d</inf> voter groups the defender defends, the outcome of the election is always different from the original one (without any attack). We show that both the OPTIMAL DEFENSE problem and the OPTIMAL ATTACK problem are computationally intractable for every scoring rule and the Condorcet voting rule even when we have only 3 candidates. We also show that the OPTIMAL DEFENSE problem for every scoring rule and the Condorcet voting rule is W[2]-hard for both the parameters k<inf>a</inf> and k<inf>d</inf>, while it admits a fixed parameter tractable algorithm parameterized by the combined parameter (k<inf>a</inf>,k<inf>d</inf>). The OPTIMAL ATTACK problem for every scoring rule and the Condorcet voting rule turns out to be much harder – it is W[1]-hard even for the combined parameter (k<inf>a</inf>,k<inf>d</inf>). We propose two greedy algorithms for the OPTIMAL DEFENSE problem and empirically show that they perform effectively on many voting profiles.