Stable Matchings with Diversity Constraints: Affirmative Action is beyond NP

IJCAI 2020, pp. 146-152, 2020.

Cited by: 0|Views37
EI
Weibo:
We identified and studied a natural, albeit highly intractable, stable matching problem enhanced with diversity constraints

Abstract:

We investigate the following many-to-one stable matching problem with diversity constraints (SMTI-Diverse): Given a set of students and a set of colleges which have preferences over each other, where the students have overlapping types, and the colleges each have a total capacity as well as quotas for individual types (the diversity con...More

Code:

Data:

Introduction
Highlights
  • Stability is a classic and central property of assignments, or matchings, of agents to each other, describing that no two agents actively prefer each other to their respective situations in the matching
  • We show that STABLE MATCHING WITH TIES AND INCOMPLETE PREFERENCES-DIVERSE can be solved in polynomial time if the number m of colleges and the maximum capacity q∞ of all colleges are constants, using a simple bruteforcing algorithm based on the following observation
  • We identified and studied a natural, albeit highly intractable, stable matching problem enhanced with diversity constraints (SMTI-DIVERSE)
  • While most of the cases are still at least NP-hard, we identified special cases for which we provided polynomial-time algorithms. This implies that the respective problems lie in the class XP in terms of parameterized complexity [Downey and Fellows, 2013; Flum and Grohe, 2006; Niedermeier, 2006; Cygan et al, 2015]
  • With respect to the number n of students, we obtain an exponential-size problem kernel, which implies that SMTIDIVERSE parameterized by n is fixed-parameter tractable, and we can show that polynomial-size kernels are unlikely to exist
  • For the number m of colleges combined with the maximum capacity q∞, our algorithms are essentially optimal because we can show that both SMI-DIVERSE and FIDIVERSE are W[2]-hard; this refutes any fixed-parameter algorithms for (m + q∞) unless FPT=W[2]
Results
  • The authors provide the algorithmic results that together allow them to complete Table 2.

    4.1 SMTI-DIVERSE with few students

    the authors deal with the case where there are few number of students.
  • The authors provide the algorithmic results that together allow them to complete Table 2.
  • 4.1 SMTI-DIVERSE with few students.
  • The authors deal with the case where there are few number of students.
  • The authors omit the superscript τ if the type vectors τ are clear from the context.
  • SMTI-DIVERSE admits a problem kernel with n students, n2 + n colleges, and 2n types, and can be solved in O(n · m · t + 2n · (2n + 1)n · n2 · t) time.
Conclusion
  • The authors identified and studied a natural, albeit highly intractable, stable matching problem enhanced with diversity constraints (SMTI-DIVERSE).
  • While most of the cases are still at least NP-hard, the authors identified special cases for which the authors provided polynomial-time algorithms.
  • This implies that the respective problems lie in the class XP in terms of parameterized complexity [Downey and Fellows, 2013; Flum and Grohe, 2006; Niedermeier, 2006; Cygan et al, 2015].
  • Summarizing, the only considered fragment left open in terms of fixed parameter tractability is SMTI-DIVERSE parameterized by (m + t)
Summary
  • Introduction:

    Stability is a classic and central property of assignments, or matchings, of agents to each other, describing that no two agents actively prefer each other to their respective situations in the matching.
  • In this work the authors investigate the notion of stability in combination with diversity, which is key in many real-world matching applications, ranging from education, through health-care systems, to job and housing markets [Abdulkadiroglu, 2005; Huang, 2010; Kamada and Kojima, 2015; Kurata et al, 2017; Ahmed et al, 2017; Benabbou et al, 2019; Gonczarowski et al, 2019; Aziz et al, 2019].
  • Results:

    The authors provide the algorithmic results that together allow them to complete Table 2.

    4.1 SMTI-DIVERSE with few students

    the authors deal with the case where there are few number of students.
  • The authors provide the algorithmic results that together allow them to complete Table 2.
  • 4.1 SMTI-DIVERSE with few students.
  • The authors deal with the case where there are few number of students.
  • The authors omit the superscript τ if the type vectors τ are clear from the context.
  • SMTI-DIVERSE admits a problem kernel with n students, n2 + n colleges, and 2n types, and can be solved in O(n · m · t + 2n · (2n + 1)n · n2 · t) time.
  • Conclusion:

    The authors identified and studied a natural, albeit highly intractable, stable matching problem enhanced with diversity constraints (SMTI-DIVERSE).
  • While most of the cases are still at least NP-hard, the authors identified special cases for which the authors provided polynomial-time algorithms.
  • This implies that the respective problems lie in the class XP in terms of parameterized complexity [Downey and Fellows, 2013; Flum and Grohe, 2006; Niedermeier, 2006; Cygan et al, 2015].
  • Summarizing, the only considered fragment left open in terms of fixed parameter tractability is SMTI-DIVERSE parameterized by (m + t)
Tables
  • Table1: Each of the students u1, . . . , u4 has preferences over the two colleges (where w1 ≻ w2 indicates that w1 is preferred to w2) and their types (Female, Local). Each of the colleges has preferences over the students, Quotas for individual types and a Capacity to accommodate students
  • Table2: A complete picture of the complexity results for FIDIVERSE and SMTI-DIVERSE (see Section 2 for the definitions). Results marked with ♦ are due to [<a class="ref-link" id="cAziz_et+al_2019_a" href="#rAziz_et+al_2019_a">Aziz et al, 2019</a>, Proposition 5.1] while the remaining ones are new. All hardness results hold even for preferences with no ties, even if the corresponding measures are upper-bounded by a constant. The NP-containment result marked with ♣ holds already when either t or q∞ is a constant. The problem variants for which fixed-parameter tractability results (FPT) exist do not admit polynomial-size kernels (see Propositions 10 and 11)
  • Table3: A description of the preference lists, quotas and capacities of the colleges, together with the preference lists of the students from {r1, r2, r3, d} ∪ X ∪ X, used in the proof of Theorem 1
  • Table4: A description of the preference lists, quotas and capacities of the colleges, together with the preference lists of the students from {r1, r2, r3, d} ∪ X ∪ X ∪ F ∪ D ∪ E, used in the proof of Proposition 2
  • Table5: The types and the preference lists of the students and the colleges are as follows where we omit “≻” to save space, let i ∈ [r] and j ∈ [s], and let [C] := c1 ≻ · · · ≻ cs, “S.”, “Pref.”, “T.”, “C.”,
Download tables as Excel
Related work
  • If the number of types is equal to one, then SMTI-DIVERSE is equivalent to the Hospitals/Residents with Lower Quotas problem where no hospital is allowed to be closed (HR-LQ2), as studied by [Hamada et al, 2016]. This problem is polynomial-time solvable when no ties are allowed [Manlove, 2013, Chapter 5.2.3]. We show that SMTI-DIVERSE becomes NP-hard even for only two types. The problem variant (HR-LQ-1) where each hospital is allowed to be closed (i.e., receive no residents) was introduced by [Biroet al., 2010] and proven to be NP-complete even if each upper and lower quota is equal to three. Our SMTI-DIVERSE is different from this problem as we do not allow colleges to be closed.

    Huang [2010] introduced the closely related CLASSIFIED STABLE MATCHING (CSM) problem, which asks for a matching that fulfills the diversity constraints and does not admit the so-called blocking coalitions. Huang showed that CSM is NP-hard and further claimed that it is in NP. However, while blocking coalitions and blocking pairs are not comparable in general, we show that our ΣP2-hardness reduction can be adapted to show that CSM is indeed beyond NP: it is also ΣP2-complete.
Funding
  • Jiehua Chen is supported by the WWTF research project (VRG18-012)
  • Robert Ganian and Thekla Hamm acknowledge support from the Austrian Science Foundation (FWF, project P31336)
Reference
  • [Abdulkadiroglu, 2005] Atila Abdulkadiroglu. College admissions with affirmative action. International Journal of Game Theory, 33(535–549), 2005. → pp. 1 and 2.
    Google ScholarLocate open access versionFindings
  • [Abraham et al., 2005] David J. Abraham, Peter Biro, and David Manlove. “Almost stable” matchings in the roommates problem. In Proceedings of the Third International Workshop on Approximation and Online Algorithms (WAOA ’05), pages 1–14, 2005. → p. 21.
    Google ScholarLocate open access versionFindings
  • [Ahmed et al., 2017] Faez Ahmed, John P. Dickerson, and Mark Fuge. Diverse weighted bipartite b-matching. In Proceedings of the 31st AAAI Conference on Artificial Intelligence (AAAI ’17), pages 35–41, 2017. → pp. 1 and 3.
    Google ScholarLocate open access versionFindings
  • [Aziz et al., 2019] Haris Aziz, Serge Gaspers, Zhaohong Sun, and Toby Walsh. From matching with diversity constraints to matching with regional quotas. In Proceedings of the 18th International Conference on Autonomous Agents and Multiagent Systems (AAMAS ’19), pages 377– 385, 2019. → pp. 1, 2, 4, 8, 12, and 25.
    Google ScholarLocate open access versionFindings
  • [Benabbou et al., 2019] Nawal Benabbou, Mithun Chakraborty, and Yair Zick. Fairness and diversity in public resource allocation problems. Bulletin of the IEEE Computer Society Technical Committee on Data Engineering IEEE, 42(3):64–75, 2019. → p. 1.
    Google ScholarLocate open access versionFindings
  • [Berman et al., 2003] Piotr Berman, Marek Karpinski, and Alex D. Scott. Approximation hardness of short symmetric instances of MAX-3SAT. Technical Report 049, ECCC, 2003. → p. 12.
    Google ScholarFindings
  • [Biroet al., 2010] Peter Biro, Tamas Fleiner, Robert W. Irving, and David Manlove. The College Admissions problem with lower and common quotas. Theoretical Computer Science, 411(34-36):3136–3153, 2010. → p. 2.
    Google ScholarLocate open access versionFindings
  • [Bredereck et al., 2014] R. Bredereck, J. Chen, S. Hartung, S. Kratsch, R. Niedermeier, O. Suchy, and G.J. Woeginger.
    Google ScholarFindings
  • A multivariate complexity analysis of lobbying in multiple referenda. Journal of Artificial Intelligence Research, 50:409–446, 2014. → p. 25.
    Google ScholarLocate open access versionFindings
  • [Bredereck et al., 2017] Robert Bredereck, Jiehua Chen, Ugo P. Finnendahl, and Rolf Niedermeier. Stable roommate with narcissistic, single-peaked, and single-crossing preferences. In Proceedings of the 5th International Conference on Algorithmic Decision Theory (ADT ’17), pages 315–330, 2017. → p. 21.
    Google ScholarLocate open access versionFindings
  • [Bredereck et al., 2018] Robert Bredereck, Piotr Faliszewski, Ayumi Igarashi, Martin Lackner, and Piotr Skowron. Multiwinner elections with diversity constraints. In Proceedings of the 32nd AAAI Conference on Artificial Intelligence (AAAI ’18), pages 933–940, 2018. → p. 3.
    Google ScholarLocate open access versionFindings
  • [Bredereck et al., 2019a] Robert Bredereck, Edith Elkind, and Ayumi Igarashi. Hedonic diversity games. In Proceedings of the 18th International Conference on Autonomous Agents and Multiagent Systems (AAMAS ’19), pages 565– 573, 2019. → p. 3.
    Google ScholarLocate open access versionFindings
  • [Bredereck et al., 2019b] Robert Bredereck, Klaus Heeger, Dusan Knop, and Rolf Niedermeier. Parameterized complexity of stable roommates with ties and incomplete lists through the lens of graph parameters. In Proceedings of the 30th International Symposium on Algorithms and Computation (ISAAC ’19), pages 44:1–44:14, 2019. → p. 21.
    Google ScholarLocate open access versionFindings
  • [Chen et al., 2018] Jiehua Chen, Danny Hermelin, Manuel Sorge, and Harel Yedidsion. How hard is it to satisfy (almost) all roommates? In Proceedings of the 45th International Colloquium on Automata, Languages, and Programming (ICALP ’18), pages 35:1–35:15, 2018. → p. 21.
    Google ScholarLocate open access versionFindings
  • [Cygan et al., 2015] Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Daniel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 20→ p. 21.
    Google ScholarLocate open access versionFindings
  • [Dom et al., 2009] Michael Dom, Daniel Lokshtanov, and Saket Saurabh. Incompressibility through colors and IDs. In Proceedings of the 36th International Colloquium on Automata, Languages, and Programming, volume 5555 of Lecture Notes in Computer Science, pages 378–389.
    Google ScholarLocate open access versionFindings
  • Springer, 2009. → p. 25.
    Google ScholarFindings
  • [Downey and Fellows, 2013] Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Springer, 2013. → pp. 21, 24, and 25.
    Google ScholarFindings
  • [Flum and Grohe, 2006] Jorg Flum and Martin Grohe. Parameterized Complexity Theory. Springer, 2006. → p. 21.
    Google ScholarFindings
  • [Garey and Johnson, 1979] Michael R. Garey and David S. Johnson. Computers and Intractability—A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, 1979. → pp. 4 and 14.
    Google ScholarLocate open access versionFindings
  • [Gonczarowski et al., 2019] Yannai A. Gonczarowski, Noam Nisan, Lior Kovalio, and Assaf Romm. Matching for the Israeli: Handling rich diversity requirements. In Proceedings of the 20th ACM Conference on Economics and Computation (ACM EC ’19), page 321, 2019. → p. 1.
    Google ScholarLocate open access versionFindings
  • [Gonzalez, 1985] Teofilo F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38:293–306, 1985. → p. 4.
    Google ScholarLocate open access versionFindings
  • [Gupta et al., 2017] Sushmita Gupta, Saket Saurabh, and Meirav Zehavi. On treewidth and stable marriage. Technical report, arXiv:1707.05404, 2017. → p. 21.
    Findings
  • [H. W. Lenstra, 1983] Jr. H. W. Lenstra. Integer programming with a fixed number of variables. Mathematics of Operations Research, 8(4):538–548, November 1983. → p. 18.
    Google ScholarLocate open access versionFindings
  • [Hafalir et al., 2013] Isa E. Hafalir, M. Bumin Yenmez, and Muhammed A. Yildirim. Effective affirmative action in school choice. Theoretical Economics, 8(2):325–363, 2013. → p. 2.
    Google ScholarLocate open access versionFindings
  • [Hamada et al., 2016] Koki Hamada, Kazuo Iwama, and Shuichi Miyazaki. The Hospitals/Residents problem with lower quotas. Algorithmica, 74(1):440–465, 2016. → p. 2.
    Google ScholarLocate open access versionFindings
  • [Heo, 2019] Eun Jeong Heo. Equity and diversity in college admissions. In The Future of Economic Design. Springer, 2019. → p. 2.
    Google ScholarFindings
  • [Huang, 2010] Chien-Chung Huang. Classified stable matching. In Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA ’10), pages 1235–1253, 2010. → pp. 1, 2, and 9.
    Google ScholarLocate open access versionFindings
  • [Inacio, 2016] Bo Inacio. Fair implementation of diversity in school choice. Games and Economic Behavior, 97:54–63, 2016. → p. 2.
    Google ScholarLocate open access versionFindings
  • [Ismaili et al., 2019] Anisse Ismaili, Naoto Hamada, Yuzhe Zhang, Takamasa Suzuki, and Makoto Yokoo. Weighted matching markets with budget constraints. Journal of Artificial Intelligence Research, 65:393–421, 2019. → p. 2.
    Google ScholarLocate open access versionFindings
  • [Kamada and Kojima, 2015] Yuichiro Kamada and Fuhito Kojima. Efficient matching under distributional constraints: Theory and applications. American Economic Review, 105(1):67–99, 2015. → p. 1.
    Google ScholarLocate open access versionFindings
  • [Kannan, 1987] Ravi Kannan. Minkowski’s convex body theorem and integer programming. Mathematics of Operations Research, 12(3):415–440, 1987. → p. 18.
    Google ScholarLocate open access versionFindings
  • [Kojima, 2019] Fuhito Kojima. New directions of study in matching with constraints. In The Future of Economic Design. Springer, 2019. → p. 2.
    Google ScholarFindings
  • [Kurata et al., 2017] Ryoji Kurata, Naoto Hamada, Atsushi Iwasaki, and Makoto Yokoo. Controlled school choice with soft bounds and overlapping types. Journal of Artificial Intelligence Research, 58:153–184, 2017. → pp. 1 and 2.
    Google ScholarLocate open access versionFindings
  • [Laslier et al., 2019] Jean-Francois Laslier, Herve Moulin, M. Remzi Sanver, and William S. Zwicker. The Future of Economic Design. Springer, 2019. → p. 2.
    Google ScholarFindings
  • [Manlove, 2013] David F. Manlove. Algorithmics of Matching Under Preferences, volume 2. WorldScientific, 2013. → pp. 1, 2, 3, 9, and 12.
    Google ScholarFindings
  • [Nguyen and Vohra, 2019] Thanh Nguyen and Rakesh Vohra. Stable matching with proportionality constraints. Operations Research, 67(6):1503–1519, 2019. → p. 2.
    Google ScholarLocate open access versionFindings
  • [Niedermeier, 2006] Rolf Niedermeier. Invitation to FixedParameter Algorithms. Oxford University Press, 2006. → p. 21.
    Google ScholarFindings
  • [Papadimitriou, 1994] Christos H. Papadimitriou. Computational Complexity. Addison-Wesley, 1994. → p. 8.
    Google ScholarFindings
  • [Stockmeyer, 1976] Larry J. Stockmeyer. The polynomialtime hierarchy. Theoretical Computer Science, 3(1):1–22, 1976. → p. 5.
    Google ScholarLocate open access versionFindings
  • [Tomoeda, 2018] Kentaro Tomoeda. Finding a stable matching under type-specific minimum quotas. Journal of Economic Theory, 176:81–117, 2018. → p. 2.
    Google ScholarLocate open access versionFindings
  • Proof. We show this by providing a parameterized reduction from the W[2]-complete SET COVER problem, parameterized by the solution size [Downey and Fellows, 2013].
    Google ScholarFindings
  • complete SET PACKING problem, parameterized by the solution size k [Downey and Fellows, 2013].
    Google ScholarFindings
  • Proof. To see this for FI-DIVERSE, we note that the reduction given by Aziz et al. [2019, Proposition 5.1] can be considered as from SET COVER. It produces an instance to FIDIVERSE where the number t of types is equal to the size of the universe U, the number of colleges is one, and the maximum capacity is equal to the set cover size k. Since SET COVER does not admit a polynomial-size problem kernel for |U |+k, it follows that neither does FI-DIVERSE admit a polynomial-size problem kernel for m + t + q∞ [Dom et al., 2009].
    Google ScholarLocate open access versionFindings
  • Proof. This follows immediately from the parameterized reduction from SET COVER to FI-DIVERSE and SMIDIVERSE given in the proof of Proposition 7, and the fact that unless NP ⊆ coNP/P, SET COVER parameterized by the size of the size of the set collection S does not admit a polynomial-size problem kernel [Bredereck et al., 2014, Theorem 5].
    Google ScholarLocate open access versionFindings
Your rating :
0

 

Tags
Comments