# Stable Matchings with Diversity Constraints: Affirmative Action is beyond NP

IJCAI 2020, pp. 146-152, 2020.

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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

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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.”,
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)
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• 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].
• complete SET PACKING problem, parameterized by the solution size k [Downey and Fellows, 2013].
• 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].
• 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].