Multiple Exchange Property for M^♮-Concave Functions and Valuated Matroids

References

• Brylawski TH (1973) Some properties of basic families of subsets. Discrete Math. 6(1):333–341.Crossref, Google Scholar
• Danilov V, Koshevoy G, Murota K (2001) Discrete convexity and equilibria in economies with indivisible goods and money. Math. Soc. Sci. 41(3):251–273.Crossref, Google Scholar
• Dress AWM, Wenzel W (1990) Valuated matroid: A new look at the greedy algorithm. Appl. Math. Lett. 3(2):33–35.Crossref, Google Scholar
• Dress AWM, Wenzel W (1992) Valuated matroids. Adv. Math. 93(2):214–250.Crossref, Google Scholar
• Fujishige S (2005) Submodular Functions and Optimization, 2nd ed. (Elsevier, Amsterdam).Google Scholar
• Fujishige S, Tamura A (2007) A two-sided discrete-concave market with possibly bounded side payments: An approach by discrete convex analysis. Math. Oper. Res. 32(1):136–155.Link, Google Scholar
• Fujishige S, Yang Z (2003) A note on Kelso and Crawford’s gross substitutes condition. Math. Oper. Res. 28(3):463–469.Link, Google Scholar
• Greene C (1973) A multiple exchange property for bases. Proc. Amer. Math. Soc. 39:45–50.Crossref, Google Scholar
• Gul F, Stacchetti E (1999) Walrasian equilibrium with gross substitutes. J. Econom. Theory 87(1):95–124.Crossref, Google Scholar
• Ikebe YT, Sekiguchi Y, Shioura A, Tamura A (2015) Stability and competitive equilibria in multi-unit trading networks with discrete concave utility functions. Japan J. Indust. Appl. Math. 32(2):373–410.Crossref, Google Scholar
• Kelso AS Jr, Crawford VP (1982) Job matching, coalition formation, and gross substitutes. Econometrica 50(6):1483–1504.Crossref, Google Scholar
• Kojima F, Tamura A, Yokoo M (2014) Designing matching mechanisms under constraints: An approach from discrete convex analysis (Extended Abstract). Proc. 7th Internat. Sympos. Algorithmic Game Theory, SAGT ’14 (Springer, Berlin), 291.Google Scholar
• Kung JPS (1986) Basis-exchange properties. White N, ed. Theory of Matroids, Chapter 4 (Cambridge University Press, London), 62–75.Crossref, Google Scholar
• McDiarmid CJH (1975) An exchange theorem for independence structures. Proc. Amer. Math. Soc. 47(2):513–514.Crossref, Google Scholar
• Murota K (1998) Fenchel-type duality for matroid valuations. Math. Programming 82(3):357–375.Crossref, Google Scholar
• Murota K (2000) Matrices and Matroids for Systems Analysis (Springer, Berlin).Google Scholar
• Murota K (2003) Discrete Convex Analysis (SIAM, Philadelphia).Crossref, Google Scholar
• Murota K (2009) Recent developments in discrete convex analysis. Cook WJ, Lovász L, Vygen J, eds. Research Trends in Combinatorial Optimization (Springer, Berlin), 219–260.Crossref, Google Scholar
• Murota K (2010) Submodular function minimization and maximization in discrete convex analysis. RIMS Kokyuroku Bessatsu B23: 193–211.Google Scholar
• Murota K (2016) Discrete convex analysis: A tool for economics and game theory. J. Mechanism and Institution Design 1(1):151–273.Crossref, Google Scholar
• Murota K, Shioura A (1999) M-convex function on generalized polymatroid. Math. Oper. Res. 24(1):95–105.Link, Google Scholar
• Murota K, Tamura A (2003) Application of M-convex submodular flow problem to mathematical economics. Japan J. Indust. Appl. Math. 20:257–277.Crossref, Google Scholar
• Schrijver A (2003) Combinatorial Optimization—Polyhedra and Efficiency (Springer, Berlin).Google Scholar
• Shioura A, Tamura A (2015) Gross substitutes condition and discrete concavity for multi-unit valuations: A survey. J. Oper. Res. Soc. Japan 58(1):61–103.Crossref, Google Scholar
• Simchi-Levi D, Chen X, Bramel J (2014) The Logic of Logistics: Theory, Algorithms, and Applications for Logistics Management, 3rd ed. (Springer, New York).Crossref, Google Scholar
• Tardos É (1985) Generalized matroids and supermodular colourings. Recski A, Lovász L, eds. Matroid Theory, Vol. 40 (Colloquia Mathematica Societatis János Bolyai, North-Holland, Amsterdam), 359–382.Google Scholar
• Woodall DR (1974) An exchange theorem for bases of matroids. J. Combin. Theory B 16(3):227–228.Crossref, Google Scholar