Legal Assignments and fast EADAM with consent via classical theory of stable matchings

Gale and Shapley's college admission problem and concept of stability (Gale and Shapley 1962) have been extensively studied, applied, and extended. In school choice problems, mechanisms often aim to obtain an assignment that is more favorable to students. We investigate two extensions introduced in this context -- legal assignments (Morrill 2016) and the EADAM algorithm (Kesten 2010) -- through the lens of classical theory of stable matchings. In any instance, the set ${\cal L}$ of legal assignments is known to contain all stable assignments. We prove that ${\cal L}$ is exactly the set of stable assignments in another instance, and that essentially any optimization problem over ${\cal L}$ can be solved within the same time bound needed for solving them over the set of stable assignments. A key tool for these results is an algorithm that finds the student-optimal legal assignment. We then generalize our algorithm to obtain the output of EADAM with any given set of consenting students without sacrificing the running time, hence improving over known algorithms in both theory and practice. Lastly, we investigate how larger the set ${\cal L}$ can be compared to the set of stable matchings in the one-to-one case, and connect legal matchings with certain concepts and open problems in the literature.