A voice for minorities: diversity in approval-based committee elections under incomplete or inaccurate information

Abstract: We study diversity in approval-based committee elections with incomplete or inaccurate information. As standard in the literature on approval-based multi-winner voting, we define diversity according to the maximum coverage problem, which is known to be NP-complete, with a best attainable polynomial time approximation ratio of $1-1/\e$. In the incomplete information model, voters can vote on only a small portion of the candidates. We suggest a greedy algorithm and a local search algorithm that query voters and use the query responses to approximate the total population's opinion. For both algorithms, we prove an upper bound on the number of queries required to get a close to $(1-1/\e)$-approximate solution with high probability. We also provide a lower bound for the query complexity of non-adaptive algorithms, that cannot adapt their querying strategy to readily obtained information. In the inaccurate information setting, voters' responses are corrupted with a probability $p\in(0,\frac{1}{2})$. We provide both an upper and a lower bound for the number of queries required to attain a $(1-1/\e)$-approximate solution with high probability. Finally, using real data from Polis, we see that our algorithms perform remarkably better than the theoretical results suggest, both with incomplete and inaccurate information.