Diverse Committees with Incomplete or Inaccurate Approval Ballots

TL;DR

This paper investigates the complexity of achieving diverse approval-based committee selections under incomplete or inaccurate voter information, proposing algorithms with near-optimal query bounds and validating their performance on real and synthetic data.

Contribution

It introduces new bounds on the number of queries needed for approximate solutions and presents algorithms that nearly match these bounds under various information settings.

Findings

Adaptive algorithms require only O(m) queries, significantly fewer than non-adaptive methods.

Greedy and local search algorithms achieve near-optimal approximation ratios.

Algorithms perform well on real and synthetic data, even with practical-sized instances.

Abstract

We study diversity in approval-based committee elections with incomplete or inaccurate information. We define diversity according to the Maximum Coverage problem, which is known to be $\mathsf{NP}$-complete, with a best attainable polynomial time approximation ratio of $1-1/e$. In the incomplete information setting, voters vote only on a small portion of the candidates, and we prove that getting arbitrarily close to the optimal approximation ratio w.h.p. requires $\Omega(m^2)$ non-adaptive queries, where $m$ is the number of candidates. This motivates studying adaptive querying algorithms, that can adapt their querying strategy to information obtained from previous query outcomes. In that setting, we lower this bound to only $\Omega(m)$ queries. We propose a greedy algorithm to match this lower bound up to log-factors. We prove the same $\tilde\Theta(m)$ bound for the generalized problem of Maximum Coverage over a matroid constraint, using a local search algorithm. Specifying a matroid of valid committees lets us implement extra structural requirements on the committee, like quota. In the inaccurate information setting, voters' responses are corrupted with a small probability. We prove $\tilde\Theta(nm)$ queries are required to attain a $(1-1/e)$-approximation with high probability, where $n$ is the number of voters. While the proven bounds show that all our algorithms are viable asymptotically, they also show that some of them would still require large numbers of queries in instances of practical relevance. Using real data from Polis as well as synthetic data, we observe that our algorithms perform well also on smaller instances, both with incomplete and inaccurate information.