Three-sided matching problem with mixed preferences

  • Feng Zhang
  • Liwei ZhongEmail author


In this paper, we study the three-sided matching problems with mixed preferences, where three agent sets are U, V and W. We discussed two matching problems with different types of preferences. The first is that each \(u\in U\) has a strict preference over set V, each \(v\in V\) has a strict preference over set W, each \(w\in W\) has a strict preference over set V and each \(w\in W\) has a strict preference over set U. The second is that each \(u\in U\) has a strict preference over set V, each \(v\in V\) has a strict preference over set W and each \(w\in W\) has a strict preference over set \(U\times V=\{(u,v)|u\in U,v\in V \}\). For these two kinds of matching problems, we give the concept of stable matching and the algorithm of solving stable matching respectively. Finally, we discuss the relationship between these two matching problems.


Three-sided matching problem Mixed preference Stable matching Algorithm 



This research is supported by the National Natural Science Foundation of China (Item Number: 71520107003), the Shanghai Science Committee of China (Item Number: 17495810503), the Gaoyuan Discipline of Shanghai-Environmental Science and Engineering (Resource Recycling Science and Engineering), and the Applied Mathematical Subject of SSPU. We would like to express our heartfelt thanks.


  1. Afacan MO (2012) Group robust stability in matching markets. Games Econ Behav 74(1):394–398MathSciNetCrossRefGoogle Scholar
  2. Anshelevich E, Bhardwaj O, Hoefer M (2017) Stable matching with network externalities. Algorithmica 78(3):1067–1106MathSciNetCrossRefGoogle Scholar
  3. Baiou M, Balinski M (2000) Many-to-many matching: stable polyandrous polygamy (or polygamous polyandry). Discrete Appl Math 101(1–3):1–12MathSciNetCrossRefGoogle Scholar
  4. Bansal V, Agrawal A, Malhotra VS (2007) Polynomial time algorithm for an optimal stable assignment with multiple partners. Theor Comput Sci 379(3):317–328MathSciNetCrossRefGoogle Scholar
  5. Biró P, McDermid E (2010) Three-sided stable matchings with cyclic preferences. Algorithmica 58(1):5–18MathSciNetCrossRefGoogle Scholar
  6. Boros E, Gurvich V, Jaslar S, Krasner D (2004) Stable matchings in three-sided systems with cyclic preferences. Discrete Math 289(1–3):1–10MathSciNetzbMATHGoogle Scholar
  7. Eriksson K, Sjöstrand J, Strimling P (2006) Three-dimensional stable matching with cyclic preferences. Math Soc Sci 52(1):77–87MathSciNetCrossRefGoogle Scholar
  8. Gale D, Shapley LS (1962) College admissions and the stability of marriage. Am Math Mon 69(1):9–15MathSciNetCrossRefGoogle Scholar
  9. Huang CC (2007) Two’s company, three’s a crowd: stable family and threesome roommates problems. In: European symposium on algorithms. Springer, Berlin, pp 558–569Google Scholar
  10. Huang CC (2010) Circular stable matching and 3-way kidney transplant. Algorithmica 58(1):137–150MathSciNetCrossRefGoogle Scholar
  11. Huang CC, Kavitha T (2015) Improved approximation algorithms for two variants of the stable marriage problem with ties. Math Program 154(1–2):353–380MathSciNetCrossRefGoogle Scholar
  12. Manlove DF (2002) The structure of stable marriage with indifference. Discrete Appl Math 122(1–3):167–181MathSciNetCrossRefGoogle Scholar
  13. Manlove DF, McBride I, Trimble J (2017) “Almost-stable” matchings in the hospitals/residents problem with couples. Constraints 22(1):50–72MathSciNetCrossRefGoogle Scholar
  14. McDermid E, Irving RW (2014) Sex-equal stable matchings: complexity and exact algorithms. Algorithmica 68(3):545–570MathSciNetCrossRefGoogle Scholar
  15. McDermid EJ, Manlove DF (2010) Keeping partners together: algorithmic results for the hospitals/residents problem with couples. J Comb Optim 19(3):279–303MathSciNetCrossRefGoogle Scholar
  16. Ng C, Hirschberg DS (1991) Three-dimensional stabl matching problems. SIAM J Discrete Math 4(2):245–252MathSciNetCrossRefGoogle Scholar
  17. Romero-Medina A (2001) ‘Sex-equal’ stable matchings. Theor Decis 50(3):197–212MathSciNetCrossRefGoogle Scholar
  18. Roth AE (1989) Two-sided matching with incomplete information about others’ preferences. Games Econ Behav 1(2):191–209MathSciNetCrossRefGoogle Scholar
  19. Zhang F, Li J, Fan J, Shen H, Shen J, Yu H (2019) Three-dimensional stable matching with hybrid preferences. J Comb Optim 37(1):330–336MathSciNetCrossRefGoogle Scholar
  20. Zhong L, Bai Y (2019) Three-sided stable matching problem with two of them as cooperative partners. J Comb Optim 37(1):286–292MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Research Center of Resource Recycling Science and EngineeringShanghai Polytechnic UniversityShanghaiChina
  2. 2.Shanghai General Hospital, School of MedicineShanghai Jiaotong UniversityShanghaiChina

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