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

In most social choice settings, the participating agents are typically required to express their preferences over the different alternatives in the form of linear orderings. While this simplifies preference elicitation, it inevitably leads to high distortion when aiming to optimize a cardinal objective such as the social welfare, since the values of the agents remain virtually unknown. A recent array of works put forward the agenda of designing mechanisms that can learn the values of the agents for a small number of alternatives via queries, and use this extra information to make a better-informed decision, 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, such as One-Sided Matching, Two-Sided Matching, General Graph Matching, and $k$-Constrained Resource Allocation. 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 that can be achieved by randomized ordinal mechanisms