A survey of approximability and inapproximability results for social welfare optimization in multiagent resource allocation

Article

Abstract

Multiagent resource allocation provides mechanisms to allocate bundles of resources to agents, where resources are assumed to be indivisible and nonshareable. A central goal is to maximize social welfare of such allocations, which can be measured in terms of the sum of utilities realized by the agents (utilitarian social welfare), in terms of their minimum (egalitarian social welfare), and in terms of their product (Nash product social welfare). Unfortunately, social welfare optimization is a computationally intractable task in many settings. We survey recent approximability and inapproximability results on social welfare optimization in multiagent resource allocation, focusing on the two most central representation forms for utility functions of agents, the bundle form and the k-additive form. In addition, we provide some new (in)approximability results on maximizing egalitarian social welfare and social welfare with respect to the Nash product when restricted to certain special cases.

Keywords

Computational social choice Multiagent resource allocation Social welfare optimization Approximability 

Mathematics Subject Classifications (2010)

68Q17 68T99 

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References

  1. 1.
    Arora, S., Lund, C.: Hardness of approximations. In: Hochbaum, D. (ed.) Approximation Algorithms for NP-Hard Problems, chap. 10, pp. 399–446. PWS Publishing Company (1996)Google Scholar
  2. 2.
    Asadpour, A., Saberi, A.: An approximation algorithm for max-min fair allocation of indivisible goods. In: Proceedings of the 39th ACM Symposium on Theory of Computing, pp. 114–121. ACM Press, New York (2007)Google Scholar
  3. 3.
    Asadpour, A., Saberi, A.: An approximation algorithm for max-min fair allocation of indivisible goods. SIAM J. Comput. 39(7), 2970–2989 (2010)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bansal, N., Sviridenko, M.: The Santa Claus problem. In: Proceedings of the 38th ACM Symposium on Theory of Computing, pp. 31–40. ACM Press, New York (2006)Google Scholar
  5. 5.
    Bezáková, I., Dani, V.: Allocating indivisible goods. SIGecom Exchanges 5(3), 11–18 (2005)CrossRefGoogle Scholar
  6. 6.
    Blumrosen, L., Nisan, N.: On the computational power of iterative auctions. In: Proceedings of the 6th ACM Conference on Electronic Commerce, pp. 29–43. ACM Press, New York (2005)Google Scholar
  7. 7.
    Blumrosen, L., Nisan, N.: Combinatorial auctions. In: Nisan, N., Roughgarden, T., Tardos, É., Vazirani, V. (eds.) Algorithmic Game Theory, chap. 11, pp. 267–299. Cambridge University Press, Cambridge (2007)CrossRefGoogle Scholar
  8. 8.
    Calinescu, G., Chekuri, C., Pál, M., Vondrák, J.: Maximizing a submodular set function subject to a matroid constraint (extended abstract). In: Proceedings of the 12th International Integer Programming and Combinatorial Optimization Conference, Lecture Notes in Computer Science, vol. 4513, pp. 182–196. Springer, Berlin Heidelberg New York (2007)Google Scholar
  9. 9.
    Chakrabarty, D., Chuzhoy, J., Khanna, S.: On allocating goods to maximize fairness. In: Proceedings of the 50th IEEE Symposium on Foundations of Computer Science, pp. 107–116. IEEE Computer Society Press, Los Alamitos (2009)Google Scholar
  10. 10.
    Chakrabarty, D., Goel, G.: On the approximability of budgeted allocations and improved lower bounds for submodular welfare maximization and GAP. In: Proceedings of the 49th IEEE Symposium on Foundations of Computer Science, pp. 687–696. IEEE Computer Society Press, Los Alamitos (2008)Google Scholar
  11. 11.
    Chekuri, C., Vondrák, J., Zenklusen, R.: Dependent randomized rounding via exchange properties of combinatorial structures. In: Proceedings of the 51st IEEE Symposium on Foundations of Computer Science, pp. 575–584. IEEE Computer Society Press, Los Alamitos (2010)Google Scholar
  12. 12.
    Chevaleyre, Y., Dunne, P., Endriss, U., Lang, J., Lemaître, M., Maudet, N., Padget, J., Phelps, S., Rodríguez-Aguilar, J., Sousa, P.: Issues in multiagent resource allocation. Informatica 30, 3–31 (2006)MATHGoogle Scholar
  13. 13.
    Chevaleyre, Y., Endriss, U., Estivie, S., Maudet, N.: Multiagent resource allocation with k-additive utility functions. In: Proceedings of the DIMACS-LAMSADE Workshop on Computer Science and Decision Theory, Annales du LAMSADE, vol. 3, pp. 83–100 (2004)Google Scholar
  14. 14.
    Chevaleyre, Y., Endriss, U., Estivie, S., Maudet, N.: Multiagent resource allocation in k-additive domains: preference representation and complexity. Ann. Oper. Res. 163, 49–62 (2008)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Csirik, J., Kellerer, H., Woeginger, G.: The exact LPT-bound for maximizing the minimum completion time. Oper. Res. Lett. 11(5), 281–287 (1992)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Deuermeyer, B., Friesen, D., Langston, M.: Scheduling to maximize the minimum processor finish time in a multiprocessor system. SIAM J. Algebr. Discrete Methods 3(2), 452–454 (1982)MathSciNetGoogle Scholar
  17. 17.
    Dobzinski, S., Nisan, N., Schapira, M.: Approximation algorithms for combinatorial auctions with complement-free bidders. Math. Oper. Res. 35(1), 1–13 (2010)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Dobzinski, S., Schapira, M.: An improved approximation algorithm for combinatorial auctions with submodular bidders. In: Proceedings of the 17th ACM-SIAM Symposium on Discrete Algorithms, pp. 1064–1073. ACM Press, New York (2006)CrossRefGoogle Scholar
  19. 19.
    Dunne, P., Wooldridge, M., Laurence, M.: The complexity of contract negotiation. Artif. Intell. 164(1–2), 23–46 (2005)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Feige, U.: On allocations that maximize fairness. In: Proceedings of the 19th ACM-SIAM Symposium on Discrete Algorithms, pp. 287–293. Society for Industrial and Applied Mathematics, Philadelphia (2008)Google Scholar
  21. 21.
    Feige, U.: On maximizing welfare when utility functions are subadditive. SIAM J. Comput. 39(1), 122–142 (2009)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Feige, U., Vondrák, J.: Approximation algorithms for allocation problems: improving the factor of 1 − 1/e. In: Proceedings of the 47th IEEE Symposium on Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos (2006)Google Scholar
  23. 23.
    Fleischer, L., Goemans, M., Mirrokni, V., Sviridenko, M.: Tight approximation algorithms for maximum general assignment problems. In: Proceedings of the 17th ACM-SIAM Symposium on Discrete Algorithms, pp. 611–620. ACM Press, New York (2006)CrossRefGoogle Scholar
  24. 24.
    Garey, M., Johnson, D.: Computers and Intractability: a Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)MATHGoogle Scholar
  25. 25.
    Gill, J.: Computational complexity of probabilistic Turing machines. SIAM J. Comput. 6(4), 675–695 (1977)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Goemans, M., Harvey, N., Iwata, S., Mirrokni, V.: Approximating submodular functions everywhere. In: Proceedings of the 20th ACM-SIAM Symposium on Discrete Algorithms, pp. 535–544. Society for Industrial and Applied Mathematics, Philadelphia (2009)Google Scholar
  27. 27.
    Golovin, D.: Max-min Fair Allocation of Indivisible Goods. Tech. Rep. CMU-CS-05-144, School of Computer Science, Carnegie Mellon University (2005)Google Scholar
  28. 28.
    Grabisch, M.: k-order additive discrete fuzzy measures and their representation. Fuzzy Sets Syst. 92(2), 167–189 (1997)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Håstad, J.: Clique is hard to approximate within n 1 − ϵ. Acta Math. 182(1), 105–142 (1999)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Håstad, J.: Some optimal inapproximability results. J. ACM 48(4), 798–859 (2001)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Hochbaum, D., Shmoys, D.: Using dual approximation algorithms for scheduling problems: theoretical and practical results. J. ACM 34(1), 144–162 (1987)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Holzman, R., Kfir-Dahav, N., Monderer, D., Tennenholtz, M.: Bundling Equilibrium in Combinatorial Auctions. Tech. Rep. cs.GT/0201010, ACM Computing Research Repository (CoRR) (2002)Google Scholar
  33. 33.
    Horowitz, E., Sahni, S.: Exact and approximate algorithms for scheduling nonidentical processors. J. ACM 23(2), 317–327 (1976)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer, Berlin Heidelberg New York (2004)CrossRefMATHGoogle Scholar
  35. 35.
    Khot, S., Kindler, G., Mossel, E., O’Donnell, R.: Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? SIAM J. Comput. 37(1), 319–357 (2007)MATHMathSciNetGoogle Scholar
  36. 36.
    Khot, S., Lipton, R., Markakis, E., Mehta, A.: Inapproximability results for combinatorial auctions with submodular utility functions. Algorithmica 52(1), 3–18 (2008)CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Khot, S., Ponnuswami, A.: Approximation algorithms for the max-min allocation problem. In: Proceedings of Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. 10th International Workshop APPROX 2007 and 11th International Workshop RANDOM 2007. Lecture Notes in Computer Science, vol. 4627, 204–217. Springer, Berlin Heidelberg New York (2007)Google Scholar
  38. 38.
    Kuhn, H.: The Hungarian method for the assignment problem. Nav. Res. Logist. Q. 2(1–2), 83–97 (1955)CrossRefGoogle Scholar
  39. 39.
    Lehmann, B., Lehmann, D., Nisan, N.: Combinatorial auctions with decreasing marginal utilities. In: Proceedings of the 3rd ACM Conference on Electronic Commerce, pp. 18–28. ACM Press, New York (2001)Google Scholar
  40. 40.
    Lehmann, D., O’Callaghan, L., Shoham, Y.: Truth revelation in approximately efficient combinatorial auctions. In: Proceedings of the 1st ACM Conference on Electronic Commerce, pp. 96–102. ACM Press, New York (1999)CrossRefGoogle Scholar
  41. 41.
    Lenstra, J., Shmoys, D., Tardos, E.: Approximation algorithms for scheduling unrelated parallel machines. Math. Program. 46(1), 259–271 (1990)CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Lipton, R., Markakis, E., Mossel, E., Saberi, A.: On approximately fair allocations of indivisible goods. In: Proceedings of the 5th ACM Conference on Electronic Commerce, pp. 125–131. ACM Press, New York (2004)Google Scholar
  43. 43.
    Lovász, L.: Submodular functions and convexity. In: Bachem, A., Grötschel, M., Korte, B. (eds.) Mathematical Programming—The State of the Art, pp. 235–257. Springer, Berlin Heidelberg New York (1983)CrossRefGoogle Scholar
  44. 44.
    Moulin, H.: Fair Division and Collective Welfare. MIT Press, Cambridge (2004)Google Scholar
  45. 45.
    Nguyen, N., Nguyen, T., Roos, M., Rothe, J.: Complexity and approximability of egalitarian and Nash product social welfare optimization in multiagent resource allocation. In: Proceedings of the 6th European Starting AI Researcher Symposium, pp. 204–215. IOS Press, Amsterdam, The Netherlands (2012)Google Scholar
  46. 46.
    Nguyen, N., Nguyen, T., Roos, M., Rothe, J.: Complexity and approximability of social welfare optimization in multiagent resource allocation. In: Brandt, F., Faliszewski, P. (eds.) Proceedings of the 4th International Workshop on Computational Social Choice, pp. 335–346. AGH University of Science and Technology, Kraków, Poland (2012)Google Scholar
  47. 47.
    Nguyen, N., Nguyen, T., Roos, M., Rothe, J.: Complexity and approximability of social welfare optimization in multiagent resource allocation (extended abstract). In: Proceedings of the 11th International Joint Conference on Autonomous Agents and Multiagent Systems, pp. 1287–1288. IFAAMAS (2012)Google Scholar
  48. 48.
    Nguyen, T., Roos, M., Rothe, J.: A survey of approximability and inapproximability results for social welfare optimization in multiagent resource allocation. In: Website Proceedings of the Special Session on Computational Social Choice at the 12th International Symposium on Artificial Intelligence and Mathematics (2012)Google Scholar
  49. 49.
    Nisan, N.: Bidding and allocation in combinatorial auctions. In: Proceedings of the 2nd ACM Conference on Electronic Commerce, pp. 1–12. ACM Press, New York (2000)Google Scholar
  50. 50.
    Papadimitriou, C.: Computational Complexity, 2nd edn. Addison-Wesley, Reading (1995)Google Scholar
  51. 51.
    Ramezani, S., Endriss, U.: Nash social welfare in multiagent resource allocation. In: Agent-Mediated Electronic Commerce. Designing Trading Strategies and Mechanisms for Electronic Markets. Lecture Notes in Business Information Processing, vol. 79, pp. 117–131. Springer, Berlin Heidelberg New York (2010)CrossRefGoogle Scholar
  52. 52.
    Roos, M., Rothe, J.: Complexity of social welfare optimization in multiagent resource allocation. In: Proceedings of the 9th International Joint Conference on Autonomous Agents and Multiagent Systems, pp. 641–648. IFAAMAS (2010)Google Scholar
  53. 53.
    Rota, G.: On the foundations of combinatorial theory I. Theory of Möbius functions. Probab. Theory Relat. Fields 2(4), 340–368 (1964)MATHMathSciNetGoogle Scholar
  54. 54.
    Rothe, J.: Complexity Theory and Cryptology. An Introduction to Cryptocomplexity. EATCS Texts in Theoretical Computer Science. Springer, Berlin Heidelberg New York (2005)Google Scholar
  55. 55.
    Rothe, J., Baumeister, D., Lindner, C., Rothe, I.: Einführung in Computational Social Choice: Individuelle Strategien und Kollektive Entscheidungen Beim Spielen, Wählen und Teilen. Spektrum Akademischer Verlag (2011)Google Scholar
  56. 56.
    Vazirani, V.: Approximation Algorithms, 2nd edn. Springer, Berlin Heidelberg New York (2003)CrossRefGoogle Scholar
  57. 57.
    Vondrák, J.: Optimal approximation for the submodular welfare problem in the value oracle model. In: Proceedings of the 40th ACM Symposium on Theory of Computing, pp. 67–74. ACM Press, New York (2008)Google Scholar
  58. 58.
    Woeginger, G.: A polynomial-time approximation scheme for maximizing the minimum machine completion time. Oper. Res. Lett. 20(4), 149–154 (1997)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Institut für InformatikHeinrich-Heine-Univ. DüsseldorfDüsseldorfGermany

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