%AFaenza, Yuri%AZhang, Xuan%BJournal Name: Operations Research
%D2022%I
%JJournal Name: Operations Research
%K
%MOSTI ID: 10317637
%PMedium: X
%TLegal Assignments and Fast EADAM with Consent via Classic Theory of Stable Matchings
%XGale and Shapley’s stable assignment problem has been extensively studied, applied, and extended. In the context of school choice, mechanisms often aim at finding an assignment that is more favorable to students. We investigate two extensions introduced in this framework—legal assignments and the efficiency adjusted deferred acceptance mechanism (EADAM) algorithm—through the lens of the classic theory of stable matchings. In any instance, the set [Formula: see text] of legal assignments is known to contain all stable assignments. We prove that [Formula: see text] is exactly the set of stable assignments in another instance. Moreover, we show that essentially all optimization problems over [Formula: see text] can be solved within the same time bound needed for solving it over the set of stable assignments. A key tool for this latter result is an algorithm that finds the student-optimal legal assignment. We then generalize our algorithm to obtain the assignment output of EADAM with any given set of consenting students without sacrificing the running time, hence largely improving in both theory and practice over known algorithms. Finally, we show that the set [Formula: see text] can be much larger than the set of stable matchings, connecting legal matchings with certain concepts and open problems in the literature.
%0Journal Article