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Computational complexity and approximability of social welfare optimization in multiagent resource allocation

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Abstract

A central task in multiagent resource allocation, which provides mechanisms to allocate (bundles of) resources to agents, is to maximize social welfare. We assume resources to be indivisible and nonshareable and agents to express their utilities over bundles of resources, where utilities can be represented in the bundle form, the \(k\)-additive form, and as straight-line programs. We study the computational complexity of social welfare optimization in multiagent resource allocation, where we consider utilitarian and egalitarian social welfare and social welfare by the Nash product. Solving some of the open problems raised by Chevaleyre et al. (2006) and confirming their conjectures, we prove that egalitarian social welfare optimization is \(\mathrm{NP}\)-complete for the bundle form, and both exact utilitarian and exact egalitarian social welfare optimization are \(\mathrm{DP}\)-complete, each for both the bundle and the \(2\)-additive form, where \(\mathrm{DP}\) is the second level of the boolean hierarchy over \(\mathrm{NP}\). In addition, we prove that social welfare optimization by the Nash product is \(\mathrm{NP}\)-complete for both the bundle and the \(1\)-additive form, and that the exact variants are \(\mathrm{DP}\)-complete for the bundle and the \(3\)-additive form. For utility functions represented as straight-line programs, we show \(\mathrm{NP}\)-completeness for egalitarian social welfare optimization and social welfare optimization by the Nash product. Finally, we show that social welfare optimization by the Nash product in the \(1\)-additive form is hard to approximate, yet we also give fully polynomial-time approximation schemes for egalitarian and Nash product social welfare optimization in the \(1\)-additive form with a fixed number of agents.

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Notes

  1. Similarly, utilitarian social welfare is characterized by independence of individual zeros of utilities: A constant shift of an agent’s utility function does not change the social welfare ordering.

  2. Every bit at a gate node is induced as usual: If \(a\) is a gate node with a \(2\)-ary boolean operation \(\sigma \), then the bit induced at \(a\) is \(b_1 \sigma b_2\), provided that \((b_1, a)\) and \((b_2, a)\) are edges of the graph, \(\sigma \) is a binary operation, and by \(b_1\) and \(b_2\) we mean the induced bits at nodes \(b_1\) and \(b_2\). For the boolean operation \(\lnot \), the definition is analogous.

  3. Namely, that the maximum utilitarian social welfare in the MARA setting to be constructed from \((S,C)\) is exactly \(q+1\) if \((S,C) \in \textsc {X3C}\) and is exactly \(q\) otherwise (see also Footnote 4).

  4. The utility of \(a_0\) for the bundle \(R\) of all resources can be set to any positive integer value in this proof. However, in the upcoming proof of Theorem 2 (which reuses and extends the present construction and uses Lemma 1), we need \(a_0\) to have a utility of exactly \(q\) for this bundle.

  5. There might be other allocations with the same utilitarian social welfare, but for no allocation the utilitarian social welfare will exceed this value.

  6. This assumption can be made without loss of generality, because we can multiply all utilities and \(K\) by their least common multiple.

  7. This implies that no agent in \(A\) bids on bundles of resources from both \(R^{(1)}\) and \(R^{(2)}\).

  8.  Since for each \(k > 1, 1\)-additive utilities can be written as \(k\)-additive utilities (namely, by setting \(u_i(T) = 0\) for all \(T \subseteq R\) with \(1 < \Vert T \Vert \le k\), see the work of Conitzer et al. [13]), \(\mathbb{Q }\textsc {-}\textsc {\!ESWO}_{{1{{-}}\mathrm{add}}}\) is a restriction of \(\mathbb{Q }\textsc {-}\textsc {\!ESWO}_{{k{{-}}\mathrm{add}}}\). Thus, proving \(\mathbb{Q }\textsc {-}\textsc {\!ESWO}_{{1{{-}}\mathrm{add}}}\,\mathrm{NP}\)-hard immediately yields \(\mathrm{NP}\)-hardness of \(\mathbb{Q }\textsc {-}\textsc {\!ESWO}_{{k{{-}}\mathrm{add}}}\) for all \(k > 1\).

  9. This approach of presenting two reductions is necessary because the value \(k\) in the \(k\)-additive representation form corresponds to the maximum vertex degree of the graph in the given \({\textsc {XIS}}\) (respectively, IS) instance, and since \({\textsc {XIS}}\) (respectively, IS) can be solved in polynomial time when this graph has a maximum vertex degree of at most two, i.e., the problem \({\textsc {XIS}}\) restricted to graphs with maximum vertex degree at most two is not \(\mathrm{DP}\)-complete and the thus restricted problem IS is not \(\mathrm{NP}\)-complete.

  10. Because we have a boolean circuit, we actually insert an edge \((x_k,o_p^q)\), where \(o_p^q, p \in \{1,\dots ,m\}, q \in \{1,2\}\), denotes the \(\vee \)-gate that is responsible for \(z_i^j\) in clause \(p\).

  11. Indeed, some results of this paper straightforwardly transfer to “elitist” social welfare; see [10] for the definition.

References

  1. Arora, S., & Lund, C. (1996). Hardness of approximations. In D. Hochbaum (Ed.), Approximation algorithms for NP-hard problems, Chap. 10 (pp. 399–446). Boston: PWS Publishing Company.

    Google Scholar 

  2. Asadpour, A., & Saberi, A. (2010). An approximation algorithm for max-min fair allocation of indivisible goods. SIAM Journal on Computing, 39(7), 2970–2989.

    Article  MATH  MathSciNet  Google Scholar 

  3. Bezáková, I., & Dani, V. (2005). Allocating indivisible goods. SIGecom Exchanges, 5(3), 11–18.

    Article  Google Scholar 

  4. Bouveret, S. (2007). Fair allocation of indivisible items: Modeling, computational complexity and algorithmics. Ph.D. thesis, Institut Supérieur de l’Aéronautique et de l’Espace, Toulouse.

  5. Bouveret, S., Lemaître, M., Fargier, H., & Lang, J. (2005). Allocation of indivisible goods: A general model and some complexity results (extended abstract). In: Proceedings of the 4th International Joint Conference on Autonomous Agents and Multiagent Systems, (pp. 1309–1310). ACM Press.

  6. Cai, J., Gunderman, T., Hartmanis, J., Hemachandra, L., Sewelson, V., Wagner, K., et al. (1988). The boolean hierarchy I: Structural properties. SIAM Journal on Computing, 16(6), 1232–1252.

    Google Scholar 

  7. Cai, J., Gundermann, T., Hartmanis, J., Hemachandra, L., Sewelson, V., Wagner, K., et al. (1989). The boolean hierarchy II: Applications. SIAM Journal on Computing, 18(1), 95–111.

    Google Scholar 

  8. Cai, J., & Meyer, G. (1987). Graph minimal uncolorability is \({\rm D}^P\)-complete. SIAM Journal on Computing, 16(2), 259–277.

    Google Scholar 

  9. Chang, R., & Kadin, J. (1995). On computing boolean connectives of characteristic functions. Theory of Computing Systems, 28(3), 173–198.

    MATH  MathSciNet  Google Scholar 

  10. Chevaleyre, Y., Dunne, P., Endriss, U., Lang, J., Lemaître, M., Maudet, N., et al. (2006). Issues in multiagent resource allocation. Informatica, 30, 3–31.

    MATH  Google Scholar 

  11. Chevaleyre, Y., Endriss, U., Estivie, S., & Maudet, N. (2004). 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).

  12. Chevaleyre, Y., Endriss, U., Estivie, S., & Maudet, N. (2008). Multiagent resource allocation in \(k\)-additive domains: Preference representation and complexity. Annals of Operations Research, 163, 49–62.

    Article  MATH  MathSciNet  Google Scholar 

  13. Conitzer, V., Sandholm, T., & Santi, P. (2005). Combinatorial auctions with \(k\)-wise dependent valuations. In: Proceedings of the 20th National Conference on Artificial Intelligence, (pp. 248–254). AAAI Press.

  14. Dunne, P., Wooldridge, M., & Laurence, M. (2005). The complexity of contract negotiation. Artificial Intelligence, 164(1–2), 23–46.

    Article  MATH  MathSciNet  Google Scholar 

  15. Garey, M., & Johnson, D. (1979). Computers and intractability: A guide to the theory of NP-completeness. New York: W. H. Freeman and Company.

    MATH  Google Scholar 

  16. Grabisch, M. (1997). \(k\)-order additive discrete fuzzy measures and their representation. Fuzzy Sets and Systems, 92(2), 167–189.

    Article  MATH  MathSciNet  Google Scholar 

  17. Håstad, J. (2001). Some optimal inapproximability results. Journal of the ACM, 48(4), 798–859.

    Article  MATH  MathSciNet  Google Scholar 

  18. Horowitz, E., & Sahni, S. (1976). Exact and approximate algorithms for scheduling nonidentical processors. Journal of the ACM, 23(2), 317–327.

    Article  MATH  MathSciNet  Google Scholar 

  19. Lehmann, D., O’Callaghan, L., & Shoham, Y. (1999). Truth revelation in approximately efficient combinatorial auctions. In: Proceedings of the 1st ACM Conference on Electronic Commerce, (pp. 96–102). ACM Press.

  20. Lipton, R., Markakis, E., Mossel, E., & Saberi, A. (2004). On approximately fair allocations of indivisible goods. In: Proceedings of the 5th ACM Conference on Electronic Commerce, (pp. 125–131). ACM Press.

  21. Moulin, H. (2004). Fair division and collective welfare. Cambridge: MIT Press.

    Google Scholar 

  22. Nguyen, N., Nguyen, T., Roos, M., & Rothe, J. (2012). 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.

  23. Nguyen, N., Nguyen, T., Roos, M., & Rothe, J. (2012). Complexity and approximability of social welfare optimization in multiagent resource allocation. In F. Brandt & P. Faliszewski (Eds.), Proceedings of the 4th International Workshop on Computational Social Choice (pp. 335–346). Kraków, Poland: AGH University of Science and Technology.

    Google Scholar 

  24. Nguyen, N., Nguyen, T., Roos, M., & Rothe, J. (2012). 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.

  25. Nguyen, N., Roos, M., & Rothe, J. (2012). Exact optimization of social welfare by the Nash product is DP-complete. In: Website Proceedings of the 12th International Symposium on Artificial Intelligence and Mathematics.

  26. Nguyen, T., Roos, M., & Rothe, J. (2012). A survey of approximability and inapproximability results for social welfare optimization in multiagent resource allocation. Annals of Mathematics and Artificial Intelligence. To appear. A preliminary version appeared in the Website Proceedings of the Special Session on Computational Social Choice at the 12th International Symposium on Artificial Intelligence and Mathematics.

  27. Nguyen, T., & Rothe, J. (2013). Envy-ratio and average Nash social welfare optimization in multiagent resource allocation (extended abstract). In: Proceedings of the 12th International Joint Conference on Autonomous Agents and Multiagent Systems. IFAAMAS. To appear.

  28. Papadimitriou, C. (1995). Computational complexity, second edn. Boston: Addison-Wesley.

    Google Scholar 

  29. Papadimitriou, C., & Wolfe, D. (1988). The complexity of facets resolved. Journal of Computer and System Sciences, 37(1), 2–13.

    Article  MATH  MathSciNet  Google Scholar 

  30. Papadimitriou, C., & Yannakakis, M. (1984). The complexity of facets (and some facets of complexity). Journal of Computer and System Sciences, 28(2), 244–259.

    Article  MATH  MathSciNet  Google Scholar 

  31. Pippenger, N., & Fischer, M. (1979). Relations among complexity measures. Journal of the ACM, 26(2), 361–381.

    Article  MATH  MathSciNet  Google Scholar 

  32. Ramezani, S., & Endriss, U. (2010). Nash social welfare in multiagent resource allocation. In: Agent-mediated electronic commerce. Designing trading strategies and mechanisms for electronic markets, (pp. 117–131). Springer, Lecture Notes in Business Information Processing No. 79.

  33. Riege, T., & Rothe, J. (2006). Completeness in the boolean hierarchy: Exact-four-colorability, minimal graph uncolorability, and exact domatic number problems—a survey. Journal of Universal Computer Science, 12(5), 551–578.

    MathSciNet  Google Scholar 

  34. Roos, M., & Rothe, J. (2010). 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.

  35. Rothe, J. (2003). Exact complexity of exact-four-colorability. Information Processing Letters, 87(1), 7–12.

    Article  MATH  MathSciNet  Google Scholar 

  36. Rothe, J. (2005). Complexity theory and cryptology. An introduction to cryptocomplexity. EATCS texts in theoretical computer science. Berlin: Springer.

    Google Scholar 

  37. Sahni, S. (1976). Algorithms for scheduling independent tasks. Journal of the ACM, 23(1), 116–127.

    Article  MATH  MathSciNet  Google Scholar 

  38. Schnorr, C. (1976). The network complexity and the Turing machine complexity of finite functions. Acta Informatica, 7(1), 95–107.

    Article  MATH  MathSciNet  Google Scholar 

  39. Vazirani, V. (2003). Approximation algorithms, second edn. Berlin: Springer.

    Book  Google Scholar 

  40. Wagner, K. (1987). More complicated questions about maxima and minima, and some closures of NP. Theoretical Computer Science, 51, 53–80.

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

We gratefully acknowledge interesting discussions with Ulle Endriss, and we thank the AAMAS-2010, AAMAS-2012, COMSOC-2012, ISAIM-2012, STAIRS-2012, and JAAMAS reviewers for their helpful comments. This work was supported in part by DFG grants RO 1202/11-1, RO 1202/12-1 (within the ESF EUROCORES program LogICCC), and RO-1202/14-1, ARC grant DP110101792, a DAAD grant for a PPP project in the PROCOPE program, and a fellowship from the Vietnamese government.

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  1. Institut für Informatik, Heinrich-Heine-Universität Düsseldorf, 40225 , Düsseldorf, Germany

    Nhan-Tam Nguyen, Trung Thanh Nguyen, Magnus Roos & Jörg Rothe

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  1. Nhan-Tam Nguyen
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Correspondence to Magnus Roos.

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Preliminary versions of parts of this paper appear in the proceedings of the 9th and the 11th International Joint Conference on Autonomous Agents and Multiagent Systems [24, 34], of the 4th International Workshop on Computational Social Choice [23], of the 6th European Starting AI Researcher Symposium [22], and of the 12th International Symposium on Artificial Intelligence and Mathematics [25].

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Nguyen, NT., Nguyen, T.T., Roos, M. et al. Computational complexity and approximability of social welfare optimization in multiagent resource allocation. Auton Agent Multi-Agent Syst 28, 256–289 (2014). https://doi.org/10.1007/s10458-013-9224-2

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