Verification of multi-layered assignment problems

The class of assignment problems is a fundamental and well-studied in the intersection Social Choice, Computational Economics Discrete Allocation. In general problem, group agents expresses preferences over set items, task to allocate items an “optimal” way. A verification variant this problem includes allocation as part input, question becomes whether “optimal”. paper, we generalize setting where each agent equipped with multiple incomplete preference lists: Each list (called layer) ranking possibly different way according criterion. particular, introduce three multi-layer problems, corresponds optimality notion that weakens global (that is, pareto layers) Informally, first requires that, for whose size exactly some input parameter k, will not be able trade their assigned among themselves benefit at least $$\alpha $$ layers; second similar, but it concerns all groups most k rather than k; third strengthens these notions by requiring larger layers. We study from perspective parameterized complexity under several natural parameterizations such number layers, agents, allocated maximum length list, more. present almost comprehensive picture respect parameters.