Book Review
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MathSciNet review:
1568045
Full text of review:
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Book Information:
Author:
S. Fujishige
Title:
Submodular functions and optimization theory
Additional book information:
Annals of Discrete Mathematics, no. 47, North Holland, Amsterdam, 270 pp., 1991, US$97.25. ISBN 0-444-88556-0.
O. N. Bondareva, Kernel theory in $n$-person games, Vestnik Leningrad. Univ. 17 (1962), no. 13, 141–142 (Russian, with English summary). MR 0149966
Imma Curiel, Giorgio Pederzoli, and Stef Tijs, Sequencing games, European J. Oper. Res. 40 (1989), no. 3, 344–351. MR 1004986, DOI 10.1016/0377-2217(89)90427-X
Jack Edmonds, Submodular functions, matroids, and certain polyhedra, Combinatorial Structures and their Applications (Proc. Calgary Internat. Conf., Calgary, Alta., 1969) Gordon and Breach, New York, 1970, pp. 69–87. MR 0270945
Jack Edmonds and Rick Giles, A min-max relation for submodular functions on graphs, Studies in integer programming (Proc. Workshop, Bonn, 1975) Ann. of Discrete Math., Vol. 1, North-Holland, Amsterdam, 1977, pp. 185–204. MR 0460169
A. Federgruen and H. Groenevelt, Characterization and optimization of achievable performance in general queueing systems, Oper. Res. 36 (1988), no. 5, 733–741. MR 967307, DOI 10.1287/opre.36.5.733
András Frank, An algorithm for submodular functions on graphs, Bonn Workshop on Combinatorial Optimization (Bonn, 1980) Ann. Discrete Math., vol. 16, North-Holland, Amsterdam-New York, 1982, pp. 97–120. MR 686302
Daniel Granot and Mehran Hojati, On cost allocation in communication networks, Networks 20 (1990), no. 2, 209–229. MR 1038723, DOI 10.1002/net.3230200207
Daniel Granot and Gur Huberman, Minimum cost spanning tree games, Math. Programming 21 (1981), no. 1, 1–18. MR 618611, DOI 10.1007/BF01584227
Daniel Granot and Gur Huberman, The relationship between convex games and minimum cost spanning tree games: a case for permutationally convex games, SIAM J. Algebraic Discrete Methods 3 (1982), no. 3, 288–292. MR 666853, DOI 10.1137/0603029
[H] Groenevelt (1985), Two algorithms for maximizing a separable concave function over a polymatroid feasible region, working paper, The Graduate School of Management, Univ. of Rochester.
Tatsuro Ichiishi, Super-modularity: applications to convex games and to the greedy algorithm for LP, J. Econom. Theory 25 (1981), no. 2, 283–286. MR 640200, DOI 10.1016/0022-0531(81)90007-7
T. Ichiishi, Comparative cooperative game theory, Internat. J. Game Theory 19 (1990), no. 2, 139–152. MR 1066786, DOI 10.1007/BF01761073
Li Qun Qi, Odd submodular functions, Dilworth functions and discrete convex functions, Math. Oper. Res. 13 (1988), no. 3, 435–446. MR 961803, DOI 10.1287/moor.13.3.435
R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
David Schmeidler, Cores of exact games. I, J. Math. Anal. Appl. 40 (1972), 214–225. MR 381720, DOI 10.1016/0022-247X(72)90045-5
[L] S. Shapley (1967), On balanced sets and cores, Naval Res. Logistics Quart. 14, 453-460.
W. W. Sharkey, Cooperative games with large cores, Internat. J. Game Theory 11 (1982), no. 3-4, 175–182. MR 692998, DOI 10.1007/BF01755727
Robert James Weber, Probabilistic values for games, The Shapley value, Cambridge Univ. Press, Cambridge, 1988, pp. 101–119. MR 989825, DOI 10.1017/CBO9780511528446.008
- [O]
- Bondareva (1962), The core of an N-person game, Vestnik Leningrad Univ. 17, 141-142. MR 0149966 (26:7450)
- [I]
- Curiel, G. Pederzoli, and S. Tijs (1989), Sequencing games, European J. Operational Res. 40, 344-351. MR 1004986 (90g:90206)
- [J]
- Edmonds (1970), Submodular functions, matroids, and certain polyhedra, Proceedings of the Calgary International Conference on Combinatorial Structures and Their Applications (R. Guy, H. Hanani, N. Sauer, and J. Schönheim, eds.), Gordon and Breach, New York. MR 0270945 (42:5828)
- [J]
- Edmonds and R. Giles (1977), A min-max relation for submodular functions on graphs, Ann. of Discrete Math. vol. 1, North-Holland, Amsterdam, 185-204. MR 0460169 (57:165)
- [A]
- Federgruen and H. Groenevelt (1988), Characterization and optimization of achievable performance in general queueing systems, Operations Res. 36, 733-741. MR 967307 (90b:60124)
- [A]
- Frank (1982), An algorithm for submodular functions on graphs, (A. Bachem, M. Grötschel, and B. Korte, eds.), Ann. of Discrete Math., vol. 16, North-Holland, Amsterdam, pp. 189-212. MR 686302 (84k:90089)
- [D]
- Granot and M. Hojati (1990), On cost allocation in communication networks, Networks 20, 209-229. MR 1038723 (90m:90117)
- [D]
- Granot and G. Huberman (1981), Minimum cost spanning tree games, Math. Programming 21, 1-18. MR 618611 (83e:90176)
- 1.
- -(1982), The relationship between convex games and minimal cost spanning tree games: a case for permutationally convex games, SIAM J. Algebra Discrete Math. 3, 288-292. MR 666853 (83i:90171)
- [H]
- Groenevelt (1985), Two algorithms for maximizing a separable concave function over a polymatroid feasible region, working paper, The Graduate School of Management, Univ. of Rochester.
- [T]
- Ichiishi (1981), Supermodularity: applications to convex games and to the greedy algorithm for LP, J. Economic Theory 25, 283-286. MR 640200 (83b:90190)
- 2.
- -(1990), Comparative cooperative game theory, Internat. J. Game Theory 19, 139-152. MR 1066786 (91m:90210)
- [L]
- Qi (1988), Odd submodular functions, Dilworth functions and discrete convex functions, Math. Oper. Res. 13, 435-446. MR 961803 (89i:90046)
- [R]
- T. Rockafellar (1970), Convex analysis, Princeton Univ. Press, Princeton, NJ. MR 0274683 (43:445)
- [D]
- Schmeidler (1972), Cores of exact games, J. Math. Anal. Appl. 40, 214-225. MR 0381720 (52:2609)
- [L]
- S. Shapley (1967), On balanced sets and cores, Naval Res. Logistics Quart. 14, 453-460.
- [W]
- W. Sharkey (1982), Cooperative games with large cores, Internat. J. Game Theory 11, 175-182. MR 692998 (84k:90112)
- [R]
- Weber (1988), Probabilistic values for games, The Shapley Value (A. Roth, ed.), Cambridge Univ. Press, Cambridge. MR 989825 (90g:90208)
Review Information:
Reviewer:
Richard P. McLean and William W. Sharkey
Journal:
Bull. Amer. Math. Soc.
29 (1993), 98-104
DOI:
https://doi.org/10.1090/S0273-0979-1993-00387-2