Abstract
Matching problems are some of the most well-studied problems in graph theory and combinatorial optimization, with a variety of theoretical as well as practical motivations. However, in many applications of optimization problems, a “solution” corresponds to real-life decisions that have major impact on humans belonging to diverse groups defined by attributes such as gender, race, or ethnicity. Due to this motivation, the notion of algorithmic fairness has recently emerged to prominence. Depending on specific application, researchers have introduced several notions of fairness.
In this paper, we study a problem called Socially Fair Matching, which combines the traditional Minimum Weight Perfect Matching problem with the notion of social fairness that has been studied in clustering literature [Abbasi et al., and Ghadiri et al., FAccT, 2021]. In our problem, the input is an edge-weighted complete bipartite graph, where the bipartition represent two groups of entities. The goal is to find a perfect matching as well as an assignment that assigns the cost of each matched edge to one of its endpoints, such that the maximum of the total cost assigned to either of the two groups is minimized.
Unlike Minimum Weight Perfect Matching, we show that Socially Fair Matching is weakly NP-hard. On the positive side, we design a deterministic PTAS for the problem when the edge weights are arbitrary. Furthermore, if the weights are integers and polynomial in the number of vertices, then we give a randomized polynomial-time algorithm that solves the problem exactly. Next, we show that this algorithm can be used to obtain a randomized FPTAS when the weights are arbitrary.
The research leading to these results has received funding from the Research Council of Norway via the project BWCA (grant no. 314528), and the European Research Council (ERC) via grant LOPPRE, reference 819416.
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Notes
- 1.
FPTAS stands for Fully Polynomial-Time Approximation Scheme, i.e., for any \(\epsilon > 0\), an algorithm that returns a \((1+\epsilon )\)-approximation in time \((n/\epsilon )^{O(1)}\).
- 2.
PTAS stands for Polynomial-Time Approximation Scheme, i.e., for any \(\epsilon > 0\), an algorithm that returns a \((1+\epsilon )\)-approximation in time \(n^{f(1/\epsilon )}\).
References
Abbasi, M., Bhaskara, A., Venkatasubramanian, S.: Fair clustering via equitable group representations. In: Elish, M.C., Isaac, W., Zemel, R.S. (eds.) FAccT 2021: 2021 ACM Conference on Fairness, Accountability, and Transparency, Virtual Event/Toronto, Canada, 3–10 March 2021, pp. 504–514. ACM (2021)
Barketau, M., Pesch, E., Shafransky, Y.: Minimizing maximum weight of subsets of a maximum matching in a bipartite graph. Discrete Appl. Math. 196, 4–19 (2015)
Berger, A., Bonifaci, V., Grandoni, F., Schäfer, G.: Budgeted matching and budgeted matroid intersection via the gasoline puzzle. Math. Program. 128(1–2), 355–372 (2011). https://doi.org/10.1007/s10107-009-0307-4
Chouldechova, A.: Fair prediction with disparate impact: a study of bias in recidivism prediction instruments. Big Data 5(2), 153–163 (2017)
Corbett-Davies, S., Pierson, E., Feller, A., Goel, S., Huq, A.: Algorithmic decision making and the cost of fairness. In: Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Halifax, NS, Canada, 13–17 August 2017, pp. 797–806. ACM (2017)
Duginov, O.: Weighted perfect matching with constraints on the total weight of its parts. J. Appl. Ind. Math. 15(3), 393–412 (2021)
Dwork, C., Hardt, M., Pitassi, T., Reingold, O., Zemel, R.: Fairness through awareness. In: Proceedings of the 3rd Innovations in Theoretical Computer Science Conference, pp. 214–226 (2012)
Dwork, C., Ilvento, C.: Group fairness under composition. In: Proceedings of the 2018 Conference on Fairness, Accountability, and Transparency (FAT* 2018) (2018)
Edmonds, J.: Paths, trees, and flowers. Can. J. Math. 17, 449–467 (1965)
Feldman, M., Friedler, S.A., Moeller, J., Scheidegger, C., Venkatasubramanian, S.: Certifying and removing disparate impact. In: Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 259–268 (2015)
García-Soriano, D., Bonchi, F.: Fair-by-design matching. Data Min. Knowl. Disc. 34(5), 1291–1335 (2020). https://doi.org/10.1007/s10618-020-00675-y
Ghadiri, M., Samadi, S., Vempala, S.S.: Socially fair k-means clustering. In: Elish, M.C., Isaac, W., Zemel, R.S. (eds.) FAccT 2021: 2021 ACM Conference on Fairness, Accountability, and Transparency, Virtual Event/Toronto, Canada, 3–10 March 2021, pp. 438–448. ACM (2021)
Huang, C., Kavitha, T., Mehlhorn, K., Michail, D.: Fair matchings and related problems. Algorithmica 74(3), 1184–1203 (2016)
Kamada, Y., Kojima, F.: Fair matching under constraints: Theory and applications (2020)
Kamishima, T., Akaho, S., Sakuma, J.: Fairness-aware learning through regularization approach. In: Spiliopoulou, M., et al. (eds.) Data Mining Workshops (ICDMW), 2011 IEEE 11th International Conference on, Vancouver, BC, Canada, 11 December 2011, pp. 643–650. IEEE Computer Society (2011)
Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations. The IBM Research Symposia Series, pp. 85–103. Springer, Boston (1972). https://doi.org/10.1007/978-1-4684-2001-2_9
Klaus, B., Klijn, F.: Procedurally fair and stable matching. Econ. Theory 27(2), 431–447 (2006)
Kleinberg, J., Mullainathan, S., Raghavan, M.: Inherent trade-offs in the fair determination of risk scores. In: 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2017)
Kress, D., Meiswinkel, S., Pesch, E.: The partitioning min-max weighted matching problem. Eur. J. Oper. Res. 247(3), 745–754 (2015)
Makarychev, Y., Vakilian, A.: Approximation algorithms for socially fair clustering. CoRR abs/2103.02512 (2021)
Papadimitriou, C.H., Yannakakis, M.: On the approximability of trade-offs and optimal access of web sources. In: 41st Annual Symposium on Foundations of Computer Science, FOCS 2000, 12–14 November 2000, Redondo Beach, California, USA, pp. 86–92. IEEE Computer Society (2000). https://doi.org/10.1109/SFCS.2000.892068
Thanh, B.L., Ruggieri, S., Turini, F.: k-NN as an implementation of situation testing for discrimination discovery and prevention. In: Apté, C., Ghosh, J., Smyth, P. (eds.) Proceedings of the 17th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Diego, CA, USA, 21–24 August 2011, pp. 502–510. ACM (2011)
Wikipedia contributors: Sister city — Wikipedia, the free encyclopedia (2022). https://en.wikipedia.org/w/index.php?title=Sister_city&oldid=1107517947. Accessed 25 Sept 2022
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Bandyapadhyay, S., Fomin, F., Inamdar, T., Panolan, F., Simonov, K. (2023). Socially Fair Matching: Exact and Approximation Algorithms. In: Morin, P., Suri, S. (eds) Algorithms and Data Structures. WADS 2023. Lecture Notes in Computer Science, vol 14079. Springer, Cham. https://doi.org/10.1007/978-3-031-38906-1_6
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