Submodular function minimization

Mathematical Programming

Volumn 112, Issue 1, 2008, Pages 45-64

  Iwata, Satoru ^a  

^a KYOTO UNIVERSITY   (Japan)

Author keywords

Algorithm; Discrete optimization; Submodular function

Indexed keywords

ALGORITHMS; FINITE DIFFERENCE METHOD; INFORMATION THEORY; OPERATIONS RESEARCH; OPTIMIZATION; QUEUEING THEORY;

ALGORITHMIC DEVOLOPMENTS; DISCRETE OPTIMIZATION; SUBMODULAR FUNCTIONS;

FUNCTION EVALUATION;

EID: 34547428890     PISSN: 00255610     EISSN: 14364646     Source Type: Journal    
DOI: 10.1007/s10107-006-0084-2     Document Type: Conference Paper

References (77)

1. Anglès d'Auriac, J.-C.: Computing the Potts free energy and submodular functions. New Optimization Algorithms in Physics. Hartmann, A.K., Rieger, H. (eds.) pp.101-117, Wiley, NewYork (2004)
2. Anglès d'Auriac J.-C., Iglói F., Preissmann M. and Sebo A. (2002). Optimal cooperation and submodularity for computing Potts' partition functions with a large number of states. J. Phys. Ser. A 35: 6973-6983
3. Bertsimas D. and Niño-Mora J. (1996). Conservation laws, extended polymatroids and multiarmed bandit problems; a polyhedral approach to indexable systems. Math. Oper. Res. 21: 257-306
4. Bixby R.E., Cunningham W.H. and Topkis D.M. (1985). Partial order of a polymatroid extreme point. Math. Oper. Res. 10: 367-378
5. Bouchet A. (1987). Greedy algorithm and symmetric matroids. Math. Program. 38: 147-159
6. Bouchet A. and Cunningham W.H. (1995). Delta-matroids, jump systems and bisubmodular polyhedra. SIAM J. Discrete Math. 8: 17-32
7. Chandrasekaran R. and Kabadi S.N. (1988). Pseudomatroids. Discrete Math. 71: 205-217
8. Mitrani I. and Coffman E.G. (1980). A characterization of waiting time performance realizable by single-server queues. Oper. Res. 28: 810-821
9. Cover T.M. (1975). A proof of the data compression theorem of Slepian and Wolf for ergodic sources. IEEE Trans. Inform. Theory IT 21: 226-228
10. Cover T.M. and Thomas J.A. (1991). Elements of Information Theory. Wiley, Newyork
11. Cunningham W.H. (1984). Testing membership in matroid polyhedra. J. Combin. Theory Ser. B 36: 161-188
12. Cunningham W.H. (1985). On submodular function minimization. Combinatorica 5: 185-192
13. Cunningham W.H. (2002). Matching, matroids and extensions. Math. Program. 91: 515-542
14. Dress A.W.M. and Havel T.F. (1986). Some combinatorial properties of discriminants in metric vector spaces. Adv. Math. 62: 285-312
15. Dress A.W. M. and Wenzel W. (1992). Valuated matroids. Adv. Math. 93: 214-250
16. Edelsbrunner H. and Guibas L.J. (1989). Topologically sweeping an arrangement. J. Comput. Syst. Sci. 38: 165-194
17. Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Guy, R., Hanani, H., Sauer, N., Schönheim, J. (eds.) Combinatorial Structures and Their Applications. Gordon and Breach (1970)
18. Edmonds J. and Karp R.M. (1972). Theoretical improvements in algorithmic efficiency for network flow problems. J. ACM 19: 248-264
19. Federgruen A. and Groenevelt H. (1988). Characterization and optimization of achievable performance in general queueing systems. Oper. Res. 36: 733-741
20. Fleiner B. (2005). Detachment of vertices of graphs preserving edge-connectivity. SIAM J. Discrete Math. 18: 581-591
21. Fleischer L. (2000). Recent progress in submodular function minimization. OPTIMA 64: 1-11
22. Fleischer, L., Iwata, S.: Improved algorithms for submodular function minimization and submodular flow. Proceedings of the 32nd ACM Symposium on Theory of Computing 107-116 (2000)
23. Fleischer L. and Iwata S. (2003). A push-relabel framework for submodular function minimization and applications to parametric optimization. Discrete Appl. Math. 131: 311-322
24. Fleischer L., Iwata S. and McCormick S.T. (2002). A faster capacity scaling algorithm for minimum cost submodular flow. Math. Programming 92: 119-139
25. Frank A. (1982). An algorithm for submodular functions on graphs. Ann. Discrete Math. 16: 97-120
26. Frank A. (1993). Submodular functions in graph theory. Discrete Math. 111: 231-241
27. Frank A. (1993). Applications of submodular functions. In: Walker, K. (eds) Surveys in Combinatorics., pp 85-136. Cambridge University Press, Cambridge
28. Fujishige S. (1978). Polymatroidal dependence structure of a set of random variables. Inform. Contr. 39: 55-72
29. Fujishige S. (1980). Lexicographically optimal base of a polymatroid with respect to a weight vector. Math. Oper. Res. 5: 186-196
30. Fujishige S. (1984). Theory of submodular programs-A Fenchel-type min-max theorem and subgradients of submodular functions. Math. Programming 29: 142-155
31. Fujishige S. (1984). Submodular systems and related topics. Math. Programming Stud. 22: 113-131
32. Fujishige S. (1997). A min-max theorem for bisubmodular polyhedra. SIAM J. Discrete Math. 10: 294-308
33. Fujishige S. (2003). Submodular function minimization and related topics. Optim. Methods Softw. 18: 169-180
34. Fujishige, S.: Submodular Functions and Optimization, Elsevier (2005)
35. Fujishige S. and Iwata S. (2006). Bisubmodular function minimization. SIAM J. Discrete Math. 19: 1065-1073
36. Fujishige S. and Zhang X. (1992). New algorithms for the intersection problem of submodular systems. Japan. J. Indust. Appl. Math. 9: 369-382
37. Gallo G., Grigoriadis M.D. and Tarjan R.E. (1989). A fast parametric network flow algorithm and applications. SIAM J. Comput. 18: 30-55
38. Goldberg A.V. and Tarjan R.E. (1988). A new approach to the maximum flow problem. J. ACM 35: 921-940
39. Grötschel M., Lovász L. and Schrijver A. (1981). The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1: 169-197
40. Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer Heidelberg (1988)
41. Hoppe B. and Tardos É (2000). The quickest transshipment problem. Math. Oper. Res. 25: 36-62
42. Itoko, T., Iwata, S.:Computational geometric approach to submodular function minimization for multiclass queueing systems. Technical Report METR 2005-29, University of Tokyo, October (2005)
43. Iwata S. (1997). A capacity scaling algorithm for convex cost submodular flows. Math. Programming 76: 299-308
44. Iwata S. (2002). A fully combinatorial algorithm for submodular function minimization. J. Combin. Theory, Ser. B 84: 203-212
45. Iwata S. (2003). A faster scaling algorithm for minimizing submodular functions. SIAM J. Comput. 32: 833-840
46. Iwata S., Fleischer L and Fujishige S. (2001). A combinatorial strongly polynomial algorithm for minimizing submodular functions. J. ACM 48: 761-777
47. Iwata S., McCormick S.T. and Shigeno M. (2005). A strongly polynomial cut canceling algorithm for minimum cost submodular flow. SIAM J. Discrete Math. 19: 304-320
48. Iwata S., Murota K. and Shigeno M. (1997). A fast parametric submodular intersection algorithm for strong map sequences. Math. Oper. Res. 22: 803-813
49. Jordán T. and Szigeti Z. (2003). Detachments preserving local edge-connectivity of graphs. SIAM J. Discrete Math. 17: 72-87
50. Khachiyan L.G. (1979). A polynomail algorithm in linear programming. Soviet Math Dokl. 20: 191-194
51. Korte B. and Vygen J. (2000). Combinatorial Optimization-Theory and Algorithms. Springer, Berlin
52. Lovász, L. Submodular functions and convexity. Mathematical Programming-The State of the Art. Bachem A., Grötschel M., Korte B.(eds.) pp.235-257 Springer, Heidelberg (1983)
53. McCormick, S.T.: Submodular function minimization. In: Aardal, K., Nemhauser, G., Weismantel, R. (eds.) Discrete Optimization, Handbooks in Operations Research, vol. 12, Elsevier (2005)
54. McCormick, S.T., Fujishige, S.: Better algorithms for bisubmodular function minimization (2005)
55. Megiddo N. (1974). Optimal flows in networks with multiple sources and sinks. Math. Programming 7: 97-107
56. Megiddo N. (1979). Combinatorial optimization with rational objective functions. Math. Oper. Res. 4: 414-424
57. Megiddo N. (1983). applying parallel computation algorithms in the design of serial algorithms. J. ACM 30: 852-865
58. Murota K. (1996). Convexity and Steinitz's exchange property. Adv. Math. 124: 272-311
59. Murota K. (1998). Discrete convex analysis. Math. Programming 83: 313-371
60. Murota, K.: Discrete Convex Analysis SIAM (2003)
61. Nagamochi H. and Ibaraki T. (1992). Computing edge-connectivity of multigraphs and capacitated graphs. SIAM J. Discrete Math. 5: 54-66
62. Nagamochi H. and Ibaraki T. (1998). A note on minimizing submodular functions. Inform. Process. Lett. 67: 239-244
63. Nagano, K.: A strongly polynomial algorithm for line search in submodular polyhedra. Technical Report METR 2004-33, University of Tokyo, June 2004
64. Nash-Williams C.St.J.A. (1985). Connected detachments of graphs and generalized Euler trails. J. London Math. Soc. 31: 17-29
65. Nash-Williams C.St.J.A. (1991). Another proof of a theorem concerning detachments of graphs. Europ. J. Combinatorics 12: 245-247
66. Nash-Williams C.St.J.A. (1995). Strongly connected mixed graphs and connected detachments of graphs. J. Combin. Math. Combin. Comput. 19: 33-47
67. Nash-Williams C.St.J.A. (1995). A direct proof of a theorem on detachments of finite graphs. J. Combin. Math. Combin. Comput. 19: 314-318
68. Queyranne M. (1993). Structure of a simple scheduling polyhedra. Math. Programming 58: 263-285
69. Queyranne M. (1998). Minimizing symmetric submodular functions. Math. Programming 82: 3-12
70. Rizzi, R.: On minimizing symmetric set functions. 20, 445-450 (2000)
71. Schrijver A. (2000). A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J. Combin. Theory Ser. B 80: 346-355
72. Schrijver A. (2003). Combinatorial Optimization-Polyhedra and Efficiency. Springer, Berlin
73. Shanthikumar J.G. and Yao D.D. (1992). Multiclass queueing systems: polymatroidal structure and optimal scheduling control. Oper. Res. 40: S293-S299
74. Shapley L.S. (1971). Cores of convex games. Int. J. Game Theory 1: 11-26
75. Slepian D. and Wolf J.K. (1973). Noiseless coding with of correlated information sources. IEEE Trans. Inform. Theory IT19: 471-480
76. Vygen J. (2003). A note on Schrijver's submodular function minimization algorithm. J. Combin. Theory Ser. B 88: 399-402
77. Whitney H. (1935). On the abstract properties of linear dependence. Amer. J. Math. 57: 509-533