Pareto optimal matchings with lower quotas

Abstract We consider the problem of allocating applicants to courses, where each applicant has a capacity, possibly greater than 1, and a subset of acceptable courses that she ranks in a strict order of preference. Each course has a lower and an upper quota, indicating that if it is assigned some applicants then their number has to be between these two bounds. We further suppose that applicants extend their preferences over courses to preferences over bundles of courses lexicographically. In this setting we present several algorithmic results concerned with the computation of Pareto optimal matchings (POMs). Firstly, we extend the Serial Dictatorship with Project Closures mechanism to the case when an applicant can be assigned more than one course. We show that unlike in the one-to-many case no mechanism is strategy-proof against dropping manipulations and that this mechanism is strategy-proof against reordering strategies only for some picking sequences. We further show the intractability of the following problems: deciding about the Pareto optimality of a given matching, computation of a POM with maximum cardinality and computation of a POM in case of indifferences.

[1]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[2]  Gerhard J. Woeginger,et al.  Group Activity Selection Problem , 2012, WINE.

[3]  S. Brams,et al.  Efficient Fair Division , 2005 .

[4]  Shuichi Miyazaki,et al.  The Hospitals/Residents Problem with Lower Quotas , 2014, Algorithmica.

[5]  David Manlove,et al.  Many-to-one Matchings with Lower Quotas: Algorithms and Complexity , 2014, ISAAC.

[6]  David Manlove,et al.  Pareto Optimal Matchings in Many-to-Many Markets with Ties , 2015, Theory of Computing Systems.

[7]  Lars-Gunnar Svensson.  Queue allocation of indivisible goods , 1994 .

[8]  Isa Emin Hafalir,et al.  Effective affirmative action in school choice , 2011 .

[9]  David Manlove,et al.  Size versus truthfulness in the house allocation problem , 2014, EC.

[10]  Mihai Manea.  Serial dictatorship and Pareto optimality , 2007, Games Econ. Behav..

[11]  J. Sethuraman,et al.  A note on object allocation under lexicographic preferences , 2014 .

[12]  David Manlove,et al.  The College Admissions problem with lower and common quotas , 2010, Theor. Comput. Sci..

[13]  David Manlove,et al.  Algorithmics of Matching Under Preferences , 2013, Bull. EATCS.

[14]  Atila Abdulkadiroglu,et al.  RANDOM SERIAL DICTATORSHIP AND THE CORE FROM RANDOM ENDOWMENTS IN HOUSE ALLOCATION PROBLEMS , 1998 .

[15]  Kate Larson,et al.  Strategyproof Quota Mechanisms for Multiple Assignment Problems , 2015, ArXiv.

[16]  P. Pattanaik,et al.  An axiomatic characterization of the lexicographic maximin extension of an ordering over a set to the power set , 1984 .

[17]  Naoyuki Kamiyama.  A note on the serial dictatorship with project closures , 2013, Oper. Res. Lett..

[18]  Katarína Cechlárová,et al.  Pareto optimality in many-to-many matching problems , 2014, Discret. Optim..

[19]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[20]  Jérôme Monnot,et al.  Optimal Reallocation under Additive and Ordinal Preferences , 2016, AAMAS.

[21]  Jérôme Lang,et al.  A General Elicitation-Free Protocol for Allocating Indivisible Goods , 2011, IJCAI.

[22]  Daniel Monte,et al.  Matching with quorums , 2013 .

[23]  Jay Sethuraman,et al.  House allocation with fractional endowments , 2011, Int. J. Game Theory.

[24]  Toby Walsh,et al.  Possible and Necessary Allocations via Sequential Mechanisms , 2014, IJCAI.

[25]  W. Bossert,et al.  Ranking Sets of Objects , 2001 .

[26]  Makoto Yokoo,et al.  Strategy-proof matching with regional minimum quotas , 2014, AAMAS.

[27]  P. Fishburn.  Axioms for Lexicographic Preferences , 1975 .

[28]  Makoto Yokoo,et al.  A Complexity Approach for Core-Selecting Exchange with Multiple Indivisible Goods under Lexicographic Preferences , 2015, AAAI.

[29]  Muhammed A. Yıldırım,et al.  School Choice with Controlled Choice Constraints: Hard Bounds Versus Soft Bounds , 2012 .

[30]  Muhammed A. Yıldırım,et al.  Effective affirmative action in school choice: Effective affirmative action in school choice , 2013 .

[31]  Kurt Mehlhorn,et al.  Pareto Optimality in House Allocation Problems , 2005, ISAAC.