The concept of M-convex function, introduced recently by Murota, is a quantitative generalization of the set of integral points in an integral base polyhedron as well as an extension of valuated matroid of Dress-Wenzel (1990). In this paper, we extend this concept to functions on generalized polymatroids with a view to providing a unified framework for efficiently solvable nonlinear discrete optimization problems. The restriction of a function to {x ∈ Z V | x(V ) = k} for k ∈ Z is called a layer. We prove the M-convexity of each layer, and reveal that the minimizers in consecutive layers are closely related. Exploiting these properties, we can solve the optimization on layers efficiently.A number of equivalent exchange axioms are given for M-convex function on generalized polymatroid.

Let G be an undirected graph with V vertices and E edges. Many algorithms have been developed for enumerating all spanning trees in G. Most of the early algorithms use a technique called`backtracking'. Recently, several algorithms using a dierent technique have been proposed by Kapoor and Ramesh (1992), Matsui (1993), and Shioura and Tamura (1993). They nd a new spanning tree by exchanging one edge of a current one. This technique has the merit of enabling us to compress the whole output of all spanning trees by outputting only relative changes of edges. Kapoor and Ramesh rst proposed an O(N+V +E) time algorithm by adopting such compact' output, where N is the number of spanning trees. Another algorithm with the same time complexity was constructed by Shioura and Tamura. These are optimal in the sense of time complexity, but not in terms of space complexity, because they take O(V E) space. We rene Shioura and Tamura's algorithm, and decrease the space complexity from O(V E) to O(V +E) while preserving time complexity. Therefore, our algorithm is optimal in the sense of both time and space complexities. 1 Introduction.

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