k-Majority digraphs and the hardness of voting with a constant number of voters

Abstract

Many hardness results in computational social choice make use of the fact that every directed graph may be induced as the pairwise majority relation of some preference profile. However, this fact requires a number of voters that is almost linear in the number of alternatives. It is therefore unclear whether these results remain intact when the number of voters is bounded, as is, for example, typically the case in search engine aggregation settings. In this paper, we provide a systematic study of majority digraphs for a constant number of voters resulting in analytical, experimental, and complexity-theoretic insights. First, we characterize the set of digraphs that can be induced by two and three voters, respectively, and give sufficient conditions for larger numbers of voters. Second, we present a surprisingly efficient implementation via SAT solving for computing the minimal number of voters that is required to induce a given digraph and experimentally evaluate how many voters are required to induce the majority digraphs of real-world and generated preference profiles. Finally, we leverage our sufficient conditions to show that the winner determination problem of various well-known voting rules remains hard even when there is only a small constant number of voters. In particular, we show that Kemeny's rule is hard to evaluate for 7 voters, while previous methods could only establish such a result for constant even numbers of voters.

Efficient, robust and effective rank aggregation for massive biological datasets

“…Works mostly focus on permutations, i.e. when the rankings to aggregate are complete and without ties [3,10,11,27,32,34,35,7,36]. More recently, a few works have addressed rank aggregation with ties [15,37] and/or aggregation of incomplete rankings [38,13,39] have also been investigated.…”

“…However, the problem of rank aggregation is well-known to be NP-hard in most cases [2,6,7]. Computing an optimal consensus ranking is currently not possible for more than a few dozens of elements.…”

Kemeny Consensus Complexity

“…But here we are mostly interested in the closeness of Slater to Kemeny, and view Theorem 10 as supporting our conjecture that Kemeny-Manipulation is Σ p 2 -complete. Many lower bound proofs for Kemeny transfer to Slater and vice versa by the following simple observation (this is implicit in any source comparing Kemeny and Slater and explicitly stated for tournaments where every arc has weight 1 in Bachmeier et al [2019]). Observation 12.…”

Kemeny Consensus Complexity

“…But here we are mostly interested in the closeness of Slater to Kemeny, to strengthen the evidence of Theorem 10 that Kemeny-Manipulation is Σ p 2 -complete. Many lower bound proofs for Kemeny transfer to Slater and vice versa by the following simple observation (this is implicit in any source comparing Kemeny and Slater and explicitly stated for tournaments where every arc has weight 1 in Bachmeier et al (2019)).…”