Abstract
We consider a general class of combinatorial optimization problems including among others allocation, multiple knapsack, matching or travelling salesman problems. The standard version of those problems is the maximum weight optimization problem where a sum of values is optimized. However, the sum is not a good aggregation function when the fairness of the distribution of those values (corresponding for example to different agents’ utilities or criteria) is important. In this paper, using the Generalized Gini Index (GGI), a well-known inequality measure, instead of the sum to model fairness, we formulate a new general problem, that we call fair combinatorial optimization. Although GGI is a non-linear aggregating function, a 0, 1-linear program (IP) can be formulated for finding a GGI-optimal solution by exploiting a linearization of GGI proposed by Ogryczak and Sliwinski [21]. However, the time spent by commercial solvers (e.g., CPLEX, Gurobi...) for solving (IP) increases very quickly with instances’ size and can reach hours even for relatively small-sized ones. As a faster alternative, we propose a heuristic for solving (IP) based on a primal-dual approach using Lagrangian decomposition. We demonstrate the efficiency of our method by evaluating it against the exact solution of (IP) by CPLEX on several fair optimization problems related to matching. The numerical results show that our method outputs in a very short time efficient solutions giving lower bounds that CPLEX may take several orders of magnitude longer to obtain. Moreover, for instances for which we know the optimal value, these solutions are quasi-optimal with optimality gap less than 0.3%.
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Notes
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Vector \(\varvec{v}\) Pareto-dominates vector \(\varvec{v}'\) if \(\forall i \in [n], v_i \ge v'_i\) and \(\exists j \in [n], v_j > v'_j\).
References
Anand, S.: The multi-criteria bipartite matching problem (2006)
Arnold, B.: Majorization and the Lorenz Order. Springer, New York (1987). https://doi.org/10.1007/978-1-4615-7379-1
Bansal, N., Sviridenko, M.: The Santa Claus problem. In: STOC, pp. 31–40 (2006)
Bezakova, I., Dani, V.: Allocating indivisible goods. ACM SIGecom Exch. 5(3), 11–18 (2005)
Chassein, A., Goerigk, M.: Alternative formulations for the ordered weighted averaging objective. Inf. Process. Lett. 115, 604–608 (2015)
Dachert, K., Gorski, J., Klamroth, K.: An augmented weighted Tchebycheff method with adaptively chosen parameters for discrete bicriteria optimization problems. Comput. Oper. Res. 39(12), 2929–2943 (2012)
Dezs, B., Juttner, A., Kovacs, P.: LEMON - an open source C++ graph template library. Electron. Notes Theor. Comput. Sci. 264(5), 23–45 (2011)
Edmonds, J.: Maximum matching and a polyhedron with 0, 1-vertices. J. Res. Natl. Bur. Stand. 69B, 125–130 (1965)
Gilbert, H., Spanjaard O.: A game-theoretic view of randomized fair multi-agent optimization. In: IJCAI Algorithmic Game Theory Workshop (2017)
Hurkala, J., Sliwinski, T.: Fair flow optimization with advanced aggregation operators in wireless mesh networks. In: Federated Conference on Computer Science and Information Systems, pp. 415–421 (2012)
Kostreva, M., Ogryczak, W., Wierzbicki, A.: Equitable aggregations and multiple criteria analysis. Eur. J. Oper. Res. 158, 362–367 (2004)
Lesca, J., Perny, P.: LP solvable models for multiagent fair allocation problems. In: ECAI (2011)
Lovész, L., Plummer, M.: Matching Theory. North Holland, Amsterdam (1986)
Luss, H.: Equitable Resource Allocation. Wiley, Hoboken (2012)
Lust, T., Teghem, J.: The multiobjective multidimensional knapsack problem: a survey and a new approach. Int. Trans. Oper. Res. 19(4), 495–520 (2012)
Moulin, H.: Axioms of Cooperative Decision Making. Cambridge University Press, Cambridge (1988)
Moulin, H.: Fair Division and Collective Welfare. MIT Press, Cambridge (2004)
Ogryczak, W., Luss, H., Pióro, M., Nace, D., Tomaszewski, A.: Fair optimization and networks: a survey. J. Appl. Math. 2014, 25 (2014)
Ogryczak, W., Perny, P., Weng, P.: On minimizing ordered weighted regrets in multiobjective Markov decision processes. In: Brafman, R.I., Roberts, F.S., Tsoukiàs, A. (eds.) ADT 2011. LNCS (LNAI), vol. 6992, pp. 190–204. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-24873-3_15
Ogryczak, W., Perny, P., Weng, P.: A compromise programming approach to multiobjective Markov decision processes. IJITDM 12, 1021–1053 (2013)
Ogryczak, W., Sliwinski, T.: On solving linear programs with the ordered weighted averaging objective. Eur. J. Oper. Res. 148, 80–91 (2003)
Ogryczak, W., Sliwinski, T., Wierzbicki, A.: Fair resource allocation schemes and network dimensioning problems. J. Telecom. Inf. Technol. 2003(3), 34–42 (2003)
Perny, P., Weng, P.: On finding compromise solutions in multiobjective Markov decision processes. In: ECAI (short paper) (2010)
Rawls, J.: The Theory of Justice. Havard University Press, Cambridge (1971)
Rodera, H., Bagajewicz, M.J., Trafalis, T.B.: Mixed-integer multiobjective process planning under uncertainty. Ind. Eng. Chem. Res. 41(16), 4075–4084 (2002)
Schrijver, A.: Theory of Linear and Integer Programming. Wiley, Chichester (1998)
Steuer, R.: Multiple Criteria Optimization. Wiley, New York (1986)
Wang, W., Lu, C.: Projection onto the capped simplex (2015). arXiv:1503.01002
Weymark, J.: Generalized Gini inequality indices. Math. Soc. Sci. 1, 409–430 (1981)
Wierzbicki, A.: A mathematical basis for satisficing decision making. Math. Model. 3, 391–405 (1982)
Yager, R.: On ordered weighted averaging aggregation operators in multi-criteria decision making. IEEE Trans. Syst. Man Cyber. 18, 183–190 (1988)
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Nguyen, V.H., Weng, P. (2017). An Efficient Primal-Dual Algorithm for Fair Combinatorial Optimization Problems. In: Gao, X., Du, H., Han, M. (eds) Combinatorial Optimization and Applications. COCOA 2017. Lecture Notes in Computer Science(), vol 10627. Springer, Cham. https://doi.org/10.1007/978-3-319-71150-8_28
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