Instance-Optimality in the Noisy Value-and Comparison-Model --- Accept, Accept, Strong Accept: Which Papers get in?

Motivated by crowdsourced computation, peer-grading, and recommendation systems, Braver- man et al. [STOC'16] recently studied the query and round complexity of classic and popu- lar problems such as finding the maximum (max), finding all elements above a certain value (threshold-v) or computing the top-k elements of an array (top-k) in a noisy environment. An illustrating example is the task of selecting papers for a conference. This task is challenging due the crowdsourcing-nature of peer reviews: (1) the results of reviews are noisy and (2) it is necessary to parallelize the review process as much as possible. Thus, we study these fundamental problems in the noisy value model and the noisy comparison model: In the noisy value model, every review returns a value (e.g. accept). In the noisy comparison model (introduced in the seminal work of Feige et al. [SICOMP'94]) a reviewer is asked a comparative yes or no question: "Is paper i better than paper j?" In a first step, we show optimal worst-case upper and lower bounds on the round vs query complexity for max and top-k in all models. For threshold-v, we obtain optimal query complexity and nearly-optimal (i.e., optimal up to a factor O(log log k), where k is the size of the output) round complexity for all models. We then go beyond the worst-case and provide--for a large range of parameters instance-optimal algorithms (w.r.t. to the query complexity)-i.e., answering the question how important knowledge of the instance is. We complement these results by showing that for some family of instances, no instance-optimal algorithm can exist. Furthermore, we show that the value-and comparison-model are for most practical settings asymptotically equivalent (for all the above mentioned problems); on the other side, the value model is strictly easier than the comparison model in the case where the papers are totally ordered.