Valuated matroid-based algorithm for submodular welfare problem
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.
KeywordsSubmodular 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.
- 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
- 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
- Azar, Y., Birnbaum, B., & Karlin, A. (2008). Improved approximation algorithms for budgeted allocations. In Automata language and programming (pp. 186–197).Google Scholar
- Beasley, J. E. (1990). ORLIB: Operations research library. http://people.brunel.ac.uk/mastjjb/jeb/info.html. Accessed November 25, 2013.
- 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
- 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
- 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
- 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
- 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
- Fujishige, S. (2005). Submodular Functions and Optimization. 2nd ed., Annals of Discrete Mathematics, vol. 58. Amsterdam: Elsevier.Google Scholar
- 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
- 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
- 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
- Murota, K. (1996a). Valuated matroid intersection, I: Optimality criteria. SIAM Journal on Discrete Mathematics, 9, 545–561.Google Scholar
- Murota, K. (1996b). Valuated matroid intersection, II: Algorithms. SIAM Journal on Discrete Mathematics, 9, 562–576.Google Scholar
- Murota, K. (1998). Discrete convex analysis. Mathematical Programming, 83, 313–371.Google Scholar
- Murota, K. (2000). Matrices and matroids for systems analysis. Berlin: Springer.Google Scholar
- Murota, K. (2010). Submodular function minimization and maximization in discrete convex analysis. RIMS Kokyuroku Bessatsu, B23, 193–211.Google Scholar
- Oxley, J. G. (1992). Matroid theory. Oxford: Oxford University Press.Google Scholar
- 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
- 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
- Wiegele, A. (2007). Biq Mac Library: A collection of Max-Cut and quadratic 0–1 programming instances of medium size. http://biqmac.uni-klu.ac.at/biqmaclib.