Taking the Final Step to a Full Dichotomy of the Possible Winner Problem in Pure Scoring Rules

The Possible Winner problem asks, given an election where the voters’ preferences over the candidates are specified only partially, whether a designated candidate can become a winner by suitably extending all the votes. Betzler and Dorn [1] proved a result that is only one step away from a full dichotomy of this problem for the important class of pure scoring rules in the case of unweighted voters and an unbounded number of candidates: Possible Winner is NP-complete for all pure scoring rules except plurality, veto, and the scoring rule with vector (2,1,…,1,0), but is solvable in polynomial time for plurality and veto. We take the final step to a full dichotomy by showing that Possible Winner is NP-complete also for the scoring rule with vector (2,1,…,1,0).

💡 Research Summary

The paper completes the long‑standing dichotomy for the Possible Winner problem under pure scoring rules with unweighted votes and an unrestricted number of candidates. The Possible Winner problem asks whether a distinguished candidate can become a winner after extending a collection of partially specified voter preferences to complete linear orders. Betzler and Dorn (2010) showed that for pure scoring rules the problem is NP‑complete in all cases except three: plurality, veto, and the rule with scoring vector (2, 1,…, 1, 0). The first two are polynomial‑time solvable, while the complexity of the third rule remained open.

The authors close this gap by proving that the rule with vector (2, 1,…, 1, 0) is also NP‑complete. Their reduction is from the classic NP‑complete HITTING SET problem. Given a hitting‑set instance (X, S, k) they construct a candidate set C that contains a distinguished candidate c, a helper candidate h, and for each element e_i∈X a family of auxiliary candidates x_i, x_i^j, y_i^j, z_i^j (where j indexes the subsets that contain e_i). The vote set V is split into a linear part V_ℓ and a partial part V_p = V_p1 ∪ V_p2 ∪ V_p3.

• V_p1 consists of k votes that rank h first and all x_i candidates last.
• V_p2 contains, for each element e_i, a carefully designed collection of votes that force specific relationships among h, x_i, the y_i^j and z_i^j candidates.
• V_p3 contains, for each subset S_j, a vote that places all x_i^j with e_i∈S_j ahead of h, with h at the bottom.

The key technical tool is the notion of a “maximum partial score” for each candidate with respect to c, introduced by Betzler and Dorn. Lemma 3.1 guarantees that, once the maximum partial scores are fixed, a polynomial‑time construction of the linear votes V_ℓ can enforce that every candidate receives exactly its maximum partial score from the partial votes, and no more.

The reduction works as follows. If the hitting‑set instance has a solution X′ of size ≤ k, the authors extend the partial votes so that each x_i with e_i∈X′ takes a last‑place position in one of the V_p1 votes, while all other x_i take a last‑place position in a vote from V_p2. This arrangement lets every auxiliary candidate achieve its prescribed maximum partial score, and guarantees that c’s total score exceeds that of any other candidate, making c a possible winner.

Conversely, suppose c is a possible winner. Because each x_i can occupy at most one last‑place position, at most k of them can be placed last in V_p1; the remaining x_i must be placed last in their corresponding V_p2 votes. The structure of V_p2 forces that any x_i placed last in V_p1 must be the only candidates that can occupy the first‑place positions in the V_p3 votes. Consequently, the set of elements whose x_i’s are last in V_p1 forms a hitting set of size ≤ k.

Thus the hitting‑set instance is a “yes’’ instance if and only if c is a possible winner, establishing NP‑hardness. Membership in NP is trivial, so the problem is NP‑complete for the (2, 1,…, 1, 0) scoring rule.

Together with the earlier results for plurality and veto, this yields a full dichotomy: for every pure scoring rule, the Possible Winner problem is polynomial‑time solvable only for plurality and veto; for all other pure scoring rules (including the (2, 1,…, 1, 0) rule) it is NP‑complete. The paper also notes immediate corollaries for related problems such as SWAP BRIBERY and unweighted coalition manipulation, which inherit NP‑hardness under the same scoring rule.