We address the following dynamic version of the school choice question: a city, named City, admits students in two temporally-separated rounds, denoted R₁ and R₂. In round R₁, the capacity of each school is fixed and mechanism M₁ finds a student optimal stable matching. In round R₂, certain parameters change, e.g., new students move into the City or the City is happy to allocate extra seats to specific schools. We study a number of Settings of this kind and give polynomial time algorithms for obtaining a stable matching for the new situations. It is well established that switching the school of a student midway, unsynchronized with her classmates, can cause traumatic effects. This fact guides us to two types of results: the first simply disallows any re-allocations in round R₂, and the second asks for a stable matching that minimizes the number of re-allocations. For the latter, we prove that the stable matchings which minimize the number of re-allocations form a sublattice of the lattice of stable matchings. Observations about incentive compatibility are woven into these results. We also give a third type of results, namely proofs of NP-hardness for a mechanism for round R₂ under certain settings.
@InProceedings{gajulapalli_et_al:LIPIcs.FSTTCS.2020.21, author = {Gajulapalli, Karthik and Liu, James A. and Mai, Tung and Vazirani, Vijay V.}, title = {{Stability-Preserving, Time-Efficient Mechanisms for School Choice in Two Rounds}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {21:1--21:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.21}, URN = {urn:nbn:de:0030-drops-132626}, doi = {10.4230/LIPIcs.FSTTCS.2020.21}, annote = {Keywords: stable matching, mechanism design, NP-Hardness} }
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