Annals of Operations Research

, Volume 229, Issue 1, pp 565–590 | Cite as

Valuated matroid-based algorithm for submodular welfare problem

  • Takanori Maehara
  • Kazuo Murota


An algorithm for the submodular welfare problem is proposed based on the theory of discrete convex analysis. The proposed algorithm is a heuristic method built upon the valuated matroid partition algorithms, and gives the exact optimal solution for a reasonable subclass of submodular welfare problems. The algorithm has a guaranteed approximation ratio for a special case. Computational results show fairly good performance of the proposed algorithm.


Submodular welfare problem Matroid Heuristic algorithm  Discrete convex analysis 



This work is supported by KAKENHI (21360045, 26280004) and the Aihara Project, the FIRST program from JSPS. The first author is supported by JST, ERATO, Kawarabayashi Project.


  1. Abrams, Z., Goel, A., & Plotkin, S. (2004). Set \(k\)-cover algorithms for energy efficient monitoring in wireless sensor networks. In Proceedings of the 3rd international symposium on information processing in sensor networks (pp. 424–432).Google Scholar
  2. Ahuja, R. K., Ergun, Ö., Orlin, J. B., & Punnen, A. P. (2002). A survey of very large-scale neighborhood search techniques. Discrete Applied Mathematics, 123, 75–102.CrossRefGoogle Scholar
  3. Andelman, N., & Mansour, Y. (2004). Auctions with budget constraints. In Proceedings of the 9th scandinavian workshop on algorithm theory (SWAT) (pp. 26–38).Google Scholar
  4. Azar, Y., Birnbaum, B., & Karlin, A. (2008). Improved approximation algorithms for budgeted allocations. In Automata language and programming (pp. 186–197).Google Scholar
  5. Beasley, J. E. (1990). ORLIB: Operations research library. Accessed November 25, 2013.
  6. Beviá, C., Quinzii, M., & Silva, J. A. (1999). Buying several indivisible goods. Mathematical Social Sciences, 37, 1–23.CrossRefGoogle Scholar
  7. 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.CrossRefGoogle Scholar
  8. Chu, P. C., & Beasley, J. E. (1997). A genetic algorithm for the generalised assignment problem. Computers and Operations Research, 24, 17–23.CrossRefGoogle Scholar
  9. 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).Google Scholar
  10. Deshpande, A., & Khuller, S. (2008). Energy efficient monitoring in sensor networks. In LATIN’ 08 proceedings of the 8th Latin American conference on theoretical informatics, pp. 436–448.Google Scholar
  11. Dress, A. W. M., & Wenzel, W. (1990). Valuated matroid: A new look at the greedy algorithm. Applied Mathematics Letters, 3, 33–35.CrossRefGoogle Scholar
  12. Dress, A. W. M., & Wenzel, W. (1992). Valuated matroids. Advances in Mathematics, 93, 214–250.CrossRefGoogle Scholar
  13. Edmonds, J. (1970). Submodular functions, matroids and certain polyhedra. In R. Guy, H. Hanani, N. Sauer, & J. Schönheim (Eds.), Combinatorial Structures and Their Applications (pp. 69–87). New York: Gordon and Breach. Also. In: Jünger & M., Reinelt, G., & Rinaldi, G. (Eds.) (2003), Combinatorial Optimization-Eureka, You Shrink! (pp. 11–26). Berlin: Springer.Google Scholar
  14. Fisher, M. L., Nemhauser, G. L., & Wolsey, L. A. (1978). An analysis of approximations for maximizing submodular set functions-II. Mathematical Programming Study, 8, 73–87.CrossRefGoogle Scholar
  15. Fleischer, L., Goemans, M. X., Mirrokni, V. S., & Sviridenko, M. (2006). Tight approximation algorithms for maximum general assignment problems. In Proceedings of the 17th annual ACM-SIAM symposium on discrete algorithm (pp. 611–620).Google Scholar
  16. Frieze, A. M., & Jerrum, M. (1995). Improved approximation algorithms for MAX \(k\)-CUT and MAX BISECTION. In Proceedings of the 4th international IPCO conference pp. 1–13. Springer: Berlin.Google Scholar
  17. Fujishige, S. (2005). Submodular Functions and Optimization. 2nd ed., Annals of Discrete Mathematics, vol. 58. Amsterdam: Elsevier.Google Scholar
  18. Fujishige, S., & Yang, Z. (2003). A note on Kelso and Crawford’s gross substitutes condition. Mathematics of Operations Research, 28, 463–469.CrossRefGoogle Scholar
  19. Garg, R., Kumar, V., & Pandit, V. (2001). Approximation algorithms for budget-constrained auctions. In APPROX’01/RANDOM’01, proceedings of the 4th international workshop on approximation algorithms for combinatorial optimization problems and 5th international workshop on randomization and approximation techniques in computer science: Approximation, random (pp. 102–113).Google Scholar
  20. Glover, F. (1996). Ejection chains, reference structures and alternating path methods for traveling salesman problems. Discrete Applied Mathematics, 65, 223–253.CrossRefGoogle Scholar
  21. Gul, F., & Stacchetti, E. (1999). Walrasian equilibrium with gross substitutes. Journal of Economic Theory, 87, 95–124.CrossRefGoogle Scholar
  22. Kelso, A. S., & Crawford, V. P. (1982). Job matching coalition formation and gross substitutes. Econometrica, 50, 1483–1504.CrossRefGoogle Scholar
  23. Lehmann, B., Lehmann, D., & Nisan, N. (2006). Combinatorial auctions with decreasing marginal utilities. Games and Economic Behavior, 55, 270–296.CrossRefGoogle Scholar
  24. Liers, F., Jünger, M., Reinelt, G., & Rinaldi, G. (2004). Computing exact ground states of hard ising spin glass problems by branch-and-cut. In A. Hartmann & H. Rieger (Eds.), New Optimization Algorithms in Physics (pp. 47–68). Berlin: Wiley-VCH.Google Scholar
  25. Lovász, L. (1983). Submodular functions and convexity. In A. Bachem, B. Korte, & M. Grötschel (Eds.), Mathematical programming: The state of the art (pp. 235–257). Berlin: Springer.CrossRefGoogle Scholar
  26. Mirrokni, V., Schapira, M., & Vondrák, J. (2008). Tight information-theoretic lower bounds for welfare maximization in combinatorial auction. ACM Conference on Electronic Commerce, 2008, 70–77.Google Scholar
  27. Murota, K. (1996a). Valuated matroid intersection, I: Optimality criteria. SIAM Journal on Discrete Mathematics, 9, 545–561.Google Scholar
  28. Murota, K. (1996b). Valuated matroid intersection, II: Algorithms. SIAM Journal on Discrete Mathematics, 9, 562–576.Google Scholar
  29. Murota, K. (1998). Discrete convex analysis. Mathematical Programming, 83, 313–371.Google Scholar
  30. Murota, K. (2000). Matrices and matroids for systems analysis. Berlin: Springer.Google Scholar
  31. Murota, K. (2003). Discrete convex analysis. Philadelphia: Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
  32. Murota, K. (2009). Recent developments in discrete convex analysis. In W. Cook, L. Lovász, & J. Vygen (Eds.), Research trends in combinatorial optimization (pp. 219–260). Berlin: Springer.CrossRefGoogle Scholar
  33. Murota, K. (2010). Submodular function minimization and maximization in discrete convex analysis. RIMS Kokyuroku Bessatsu, B23, 193–211.Google Scholar
  34. Oxley, J. G. (1992). Matroid theory. Oxford: Oxford University Press.Google Scholar
  35. Shioura, A. (2012). Matroid rank functions and discrete concavity. Japan Journal of Industrial and Applied Mathematics, 29, 535–546.CrossRefGoogle Scholar
  36. Shioura, A., & Suzuki, S. (2012). Optimal allocation problem with quadratic utility functions and its relationship with graph cut. Journal of Operations Research Society of Japan, 55, 92–105.Google Scholar
  37. Slijepcevic, S. & Potkonjak, M. (2001). Power efficient organization of wireless sensor networks. In Proceeding of IEEE international conference on communications 2001 (pp. 472–476).Google Scholar
  38. Srinivasan, A. (2008). Budgeted allocations in the full-information setting. In M. Goemans, K. Jansen, J. D. P. Rolim, & L. Trevisan (Eds.), Approximation, randomization and combinatorial optimization: Algorithms and techniques (pp. 247–253). Berlin: Springer.CrossRefGoogle Scholar
  39. Wiegele, A. (2007). Biq Mac Library: A collection of Max-Cut and quadratic 0–1 programming instances of medium size.

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Mathematics and Systems EngineeringShizuoka UniversityShizuokaJapan
  2. 2.Department of Mathematical InformaticsUniversity of TokyoTokyoJapan

Personalised recommendations