Don't Roll the Dice, Ask Twice: The Two-Query Distortion of Matching Problems and Beyond

In most social choice settings, the participating agents express their preferences over the different alternatives in the form of linear orderings. While this clearly simplifies preference elicitation, it inevitably leads to poor performance with respect to optimizing a cardinal objective, such as the social welfare, since the values of the agents remain virtually unknown. This loss in performance because of lack of information is measured by the notion of distortion. A recent array of works put forward the agenda of designing mechanisms that learn the values of the agents for a small number of alternatives via queries, and use this limited extra information to make better-informed decisions, thus improving distortion. Following this agenda, in this work we focus on a class of combinatorial problems that includes most well-known matching problems and several of their generalizations. For problems such as One-Sided Matching, Two-Sided Matching, General Graph Matching, and Short Cycle Packing, we design two-query mechanisms that achieve the best-possible worst-case distortion in terms of social welfare, and outperform the best-possible expected distortion achieved by randomized ordinal mechanisms. Our results extend to problems like k-Constrained Resource Allocation, General Graph k-Matching, and k-Clique Packing, when k is restricted to be any constant.