Parameterized Control Complexity in Fallback Voting

We study the parameterized control complexity of fallback voting, a voting system that combines preference-based with approval voting. Electoral control is one of many different ways for an external agent to tamper with the outcome of an election. We show that adding and deleting candidates in fallback voting are W[2]-hard for both the constructive and destructive case, parameterized by the amount of action taken by the external agent. Furthermore, we show that adding and deleting voters in fallback voting are W[2]-hard for the constructive case, parameterized by the amount of action taken by the external agent, and are in FPT for the destructive case.

💡 Research Summary

This paper investigates the parameterized control complexity of fallback voting (FV), a hybrid voting system that merges preference rankings with approval voting. In FV each voter supplies an approval set Sᵥ ⊆ C together with a strict linear order of the approved candidates. Winners are determined level by level: at level i the candidates that receive more than half of the voters’ approvals among the top‑i approved candidates are examined; if a unique candidate exceeds the majority it wins, otherwise the process falls back to the next level. If no level yields a winner, the candidate(s) with the highest total approval win.

The authors focus on four classic types of electoral control: adding candidates, deleting candidates, adding voters, and deleting voters. For each type they consider both constructive control (the chair wants a distinguished candidate c to become the unique winner) and destructive control (the chair wants c not to be a winner). The parameter k denotes the maximum number of candidates or voters the chair may add or delete. The central question is whether the control problem is fixed‑parameter tractable (FPT) with respect to k, or whether it belongs to higher levels of the W‑hierarchy, in particular W