[Paper Review] On the Exact Amount of Missing Information that makes Finding Possible Winners Hard

Palash Dey, Neeldhara Misra|arXiv (Cornell University)|Oct 26, 2016

Game Theory and Voting Systems4 citations

TL;DR

This paper identifies the exact threshold of missing pairwise comparisons in partial votes that renders the Possible Winner problem NP-complete for a broad class of voting rules, including scoring rules (e.g., Borda, plurality), Copelandα, maximin, and Bucklin. It proves that for many of these rules, the problem remains NP-complete even with just one undetermined pair per vote, while it becomes polynomial-time solvable when no such pairs exist, establishing tight computational boundaries for incomplete information in voting systems.

ABSTRACT

We consider election scenarios with incomplete information, a situation that arises often in practice. There are several models of incomplete information and accordingly, different notions of outcomes of such elections. In one well-studied model of incompleteness, the votes are given by partial orders over the candidates. In this context we can frame the problem of finding a possible winner, which involves determining whether a given candidate wins in at least one completion of a given set of partial votes for a specific voting rule. The possible winner problem is well-known to be NP-complete in general, and it is in fact known to be NP-complete for several voting rules where the number of undetermined pairs in every vote is bounded only by some constant. In this paper, we address the question of determining precisely the smallest number of undetermined pairs for which the possible winner problem remains NP-complete. In particular, we find the exact values of $t$ for which the possible winner problem transitions to being NP-complete from being in P, where $t$ is the maximum number of undetermined pairs in every vote. We demonstrate tight results for a broad subclass of scoring rules which includes all the commonly used scoring rules (such as plurality, veto, Borda, $k$-approval, and so on), Copeland$^α$ for every $α\in[0,1]$, maximin, and Bucklin voting rules. A somewhat surprising aspect of our results is that for many of these rules, the possible winner problem turns out to be hard even if every vote has at most one undetermined pair of candidates.

Motivation & Objective

• To determine the precise number of undetermined pairwise comparisons per vote that causes the Possible Winner problem to become NP-complete for common voting rules.
• To establish tight computational boundaries—specifically, the minimal threshold of missing information—beyond which the problem transitions from being in P to NP-complete.
• To resolve the complexity dichotomy for a wide class of voting rules by characterizing the exact value of t (maximum undetermined pairs per vote) at which NP-completeness emerges.
• To provide a complete classification of the Possible Winner problem's complexity based on the number of missing pairwise comparisons in partial votes.

Proposed method

• The authors analyze the structure of partial votes with t undetermined pairs and construct reductions from known NP-complete problems, such as 3-Dimensional Matching, to prove hardness.
• They define a class of 'smooth' scoring rules where score vectors for m+1 candidates are derived from those of m candidates via duplication or extension at endpoints.
• For each voting rule, they perform case analysis based on the score vector's structure—particularly the presence of patterns like (α+1,α+1,α) or (α+2,α+1,α+1,α)—to determine the threshold t.
• They use max-flow techniques and combinatorial arguments to model candidate rankings and verify that the distinguished candidate c can win only under specific completions of partial votes.
• They design a transformation from a 3-Dimensional Matching instance to a Possible Winner instance with controlled missing information, preserving equivalence.
• They prove polynomial-time solvability when t ≤ 1 for certain rules by leveraging a k-approval algorithm on a carefully constructed extension where c is prioritized.

Experimental results

Research questions

• RQ1 What is the smallest number of undetermined pairwise comparisons per vote that makes the Possible Winner problem NP-complete for scoring rules?
• RQ2 For which classes of scoring rules does the problem remain NP-complete even when each vote contains only one undetermined pair?
• RQ3 Does the complexity threshold depend on the structure of the score vector, such as repeated or consecutive score differences?
• RQ4 Can the NP-completeness of the Possible Winner problem be established with just one missing pairwise comparison for Copelandα, maximin, and Bucklin voting rules?
• RQ5 Is there a precise t such that the problem is in P for t ≤ t₀ and NP-complete for t ≥ t₀ + 1 across all considered voting rules?

Key findings

• The Possible Winner problem is NP-complete for all scoring rules with two distinct nonzero differences between consecutive scores—even when each vote has at most one undetermined pair (t ≤ 1).
• For scoring rules containing the pattern (α+1, α+1, α), the problem is NP-complete when t ≥ 2 and in P when t ≤ 1.
• For scoring rules with the pattern (α+2, α+1, α+1, α), the problem is NP-complete when t ≥ 3 and in P when t ≤ 2.
• For k-approval and k-veto rules (k > 1), the problem is NP-complete when t ≥ 4 and in P when t ≤ 3.
• For the scoring rule (2,1,1,…,1,0), the problem is NP-complete when t ≥ m−1 and in P when t ≤ m−2.
• For Copelandα, maximin, and Bucklin voting rules, the problem is NP-complete even when t ≤ 2, indicating that just two missing comparisons suffice to induce hardness.