The Complexity of Determining the Necessary and Possible Top-k Winners in Partial Voting Profiles

💡 Research Summary

This paper investigates the computational complexity of determining necessary and possible top‑k winners in elections where voters’ preferences are only partially known. Building on the framework introduced by Konczak and Lang (2005) for necessary winners (candidates who win in every completion of a partial profile) and possible winners (candidates who win in at least one completion), the authors extend the notion to the top‑k setting: a candidate is a necessary top‑k winner if it appears among the k highest‑scoring candidates (according to a given tie‑breaking order) in every completion, and a possible top‑k winner if it does so in some completion.

The study focuses on positional scoring rules, a broad family that includes plurality, veto, t‑approval, t‑veto, and the Borda rule. The authors first recall the known classification for the single‑winner case: necessary winner (NW) is polynomial‑time solvable for all positional rules, while possible winner (PW) is NP‑complete for every pure scoring rule except plurality and veto, where PW is in P.

The main contributions are twofold:

1. Complexity when k is part of the input.

   • For every pure positional scoring rule, the possible top‑k winner problem (PTW) is NP‑complete. This holds even for plurality and veto, which are the only rules where PW is easy in the single‑winner case; the hardness emerges as soon as k > 1. The proof uses reductions from classic NP‑complete problems: Exact Cover by 3‑Sets (X3C) for plurality and Dominating Set for the general case.
   • For the necessary top‑k winner problem (NTW), the authors show coNP‑completeness for plurality and veto, and extend the result to a wide class of binary scoring rules (including t‑approval, t‑veto, and Borda) by introducing the complement‑reversed scoring rule rᴿ. They prove a tight correspondence: NTW for a rule r reduces to the complement of PTW for rᴿ, and vice‑versa. Consequently, NTW is coNP‑complete for veto as well.

2. Complexity when k is fixed (a constant).

   • If the scores produced by the rule are polynomially bounded in the number of candidates, NTW becomes polynomial‑time solvable for any positional rule. The algorithm enumerates feasible score intervals and checks whether a candidate can be forced out of the top‑k in any completion, which can be expressed as a linear‑programming or flow problem.
   • For plurality and veto, PTW is also polynomial‑time solvable when k is a constant, because the limited number of points a candidate can receive allows a direct enumeration of all possible completions that could place the candidate inside the top‑k.

The paper also discusses the relationship between top‑k winners and multi‑winner (committee) selection. Top‑k winners can be seen as a special case of committee selection without additional constraints (e.g., proportionality). The authors note that determining necessary or possible committee members under incomplete preferences remains an open and challenging direction, and their results imply tractability for certain Condorcet‑type committees under plurality and veto.

Methodologically, the work combines classic NP‑hardness reductions with a novel symmetry argument based on score complementarity. The reductions are carefully constructed: for NTW under plurality, each element of an X3C instance corresponds to a voter who can only vote for edges covering that element, and a distinguished candidate c* must be outranked by exactly q edges iff an exact cover exists. For PTW, the Dominating Set reduction forces a candidate to be a possible top‑k winner precisely when a small dominating set exists.

In summary, the paper establishes a clear dichotomy:

• When k is part of the input: PTW is NP‑complete for all pure positional rules; NTW is coNP‑complete for a broad class that includes plurality, veto, and many binary rules.
• When k is fixed: Both NTW and PTW become tractable for rules with polynomially bounded scores, and specifically for plurality and veto.

These findings highlight that the computational difficulty of winner determination escalates dramatically when moving from a single winner to a top‑k set, especially when k is not bounded a priori. The complement‑reversed transformation offers a powerful tool for relating necessary and possible winner problems across different scoring rules, and may find applications in other areas of computational social choice. The work thus provides both a comprehensive complexity map for top‑k winner problems under partial information and a foundation for future research on multi‑winner elections with incomplete preferences.