Constant distortion with one-round linear comparison queries

Establish whether a voting protocol can achieve a constant metric distortion bound using a single round of $\mathcal{O}(m)$ pairwise comparison queries per voter.

Background

The paper shows that its $k$-Plurality Approval protocol can be implemented using $\mathcal{O}(m)$ pairwise comparisons per voter, but requires two rounds of interaction. The queries within each round can be posed independently of the queries made to other voters in that round.

The unresolved issue is whether the same asymptotic query complexity and a constant metric distortion guarantee are possible in a single round. The paper notes that the best currently known distortion bound for this one-round setting is $\mathcal{O}(\log(m))$.