Abstract M-convex functions, introduced by Murota (1996, 1998), enjoy various desirable properties as "discrete convex functions. " In this paper, we propose two new polynomial-time scaling algorithms for the minimization of an M-convex function. Both algorithms apply a scaling technique to a greedy algorithm for M-convex function minimization, and run as fast as the previous minimization algorithms. We also specialize our scaling algorithms for the resource allocation problem which is a special case of M-convex function minimization.

The concept of M-convex functions plays a central role in "discrete convex analysis," a unified framework of discrete optimization recently developed by Murota and others. This paper gives two new characterizations of M/M " -convex functions generalizing Gul and Stacchetti's results on the equivalence among the single improvement condition, the gross substitutes condition and the no complementarities condition for set functions (utility functions on f0; 1g vectors) as well as Fujishige and Yang's observation on the connection to M-convexity. We also discuss implications of our results in an exchange economy with indivisible goods. Kazuo Murota Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan. phone: +81-75-753-7221, facsimile: +81-75-753-7272 e-mail: murota@kurims.kyoto-u.ac.jp. Akihisa Tamura Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan. phone: +81-75-753-7236, facsimile: +81-75-753-7272 e-mail: tamura@ku...