Abstract
We consider the problem of minimizing a submodular function f defined on a set V with n elements. We give a combinatorial algorithm that runs in O(n 5 EO + n 6) time, where EO is the time to evaluate f(S) for some S ⊆ V. This improves the previous best strongly polynomial running time by more than a factor of n.
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Keywords
- Distance Function
- Combinatorial Algorithm
- Submodular Function
- Iterative Local Search
- Ellipsoid Algorithm
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References
Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia (1994)
Cunningham, W.H.: On Submodular Function Minimization. Combinatorica 3, 185–192 (1985)
Edmonds, J.: Submodular Functions, Matroids, and Certain Polyhedra. In: Guy, R., Hanani, H., Sauer, N., Schönheim, J. (eds.) Combinatorial Structures and their Applications, pp. 69–87. Gordon and Breach, New York (1970)
Fleischer, L.K.: Recent Progress in Submodular Function Minimization. Optima, 1–11 (2000)
Fleischer, L.K., Iwata, S.: Improved Algorithms for Submodular Function Minimization and Submodular Flow. In: Proceedings of the 32th Annual ACM Symposium on Theory of Computing, pp. 107–116 (2000)
Fujishige, S.: Submodular Functions and Optimization, 2nd edn. North-Holland, Amsterdam (2005)
The Ellipsoid Algorithm and its Consequences in Combinatorial Optimization. Combinatorica 1, 499–513 (1981)
Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Heidelberg (1988)
Iwata, S.: A Faster Scaling Algorithm for Minimizing Submodular Functions. SIAM J. on Computing 32, 833–840 (2002)
Iwata, S., Fleischer, L., Fujishige, S.: A Combinatorial, Strongly Polynomial-Time Algorithm for Minimizing Submodular Functions. J. ACM 48, 761–777 (2001)
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, Heidelberg (1983)
McCormick, S.T.: Submodular Function Minimization. In: Aardal, K., Nemhauser, G., Weismantel, R. (eds.) Handbooks in Operations Research and Management Science, vol. 12, Elsevier, Amsterdam (2005)
Schrijver, A.: A Combinatorial Algorithm Minimizing Submodular Functions in Strongly Polynomial Time. J. Combin. Theory Ser. B 80, 346–355 (2000)
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin (2003)
Vygen, J.: A Note on Schrijver’s Submodular Function Minimization Algorithm. Journal of Combinatorial Theory B 88, 399–402 (2003)
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Orlin, J.B. (2007). A Faster Strongly Polynomial Time Algorithm for Submodular Function Minimization. In: Fischetti, M., Williamson, D.P. (eds) Integer Programming and Combinatorial Optimization. IPCO 2007. Lecture Notes in Computer Science, vol 4513. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72792-7_19
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DOI: https://doi.org/10.1007/978-3-540-72792-7_19
Publisher Name: Springer, Berlin, Heidelberg
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