The concept of M-convex function gives a unified framework for discrete optimization problems with nonlinear objective functions such as the minimum convex cost flow problem and the convex resource allocation problem. M-convex function minimization is one of the most fundamental problems concerning Mconvex functions. It is known that a minimizer of an M-convex function can be found by a steepest descent algorithm in a finite number of iterations. Recently, the exact number of iterations required by a basic steepest descent algorithm was obtained. Furthermore, it was shown that the trajectory of the solutions generated by the basic steepest descent algorithm is a geodesic between the initial solution and the nearest minimizer. In this paper, we give a simpler and shorter proof of this claim by refining the minimizer cut property. We also consider the minimization of a jump M-convex function, which is a generalization of M-convex function, and analyze the number of iterations required by the basic steepest descent algorithm. In particular, we show that the trajectory of the solutions generated by the algorithm is a geodesic between the initial solution and the nearest minimizer.

The concept of jump system, introduced by Bouchet and Cunningham (1995), is a set of integer points satisfying a certain exchange property. We consider the minimization of a separable convex function on a jump system. It is known that the problem can be solved by a greedy algorithm. In this paper, we are interested in whether the greedy algorithm has the geodesic property, which means that the trajectory of solution generated by the algorithm is a geodesic from the initial solution to a nearest optimal solution. We show that a special implementation of the greedy algorithm enjoys the geodesic property, while the original algorithm does not. As a corollary to this, we present a new greedy algorithm for linear optimization on a delta-matroid and show that the algorithm has the geodesic property.

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Separable convex resource allocation problem aims at finding an allocation of a discrete resource to several activities that minimizes a separable convex function representing the total cost or the total loss. In this paper, we consider the separable convex resource allocation problem with an additional constraint that the L 1 -distance between a given vector and a feasible solution is bounded by a given positive constant. We prove that the simplest separable convex resource allocation problem with the L 1 -distance constraint can be reformulated as a submodular resource allocation problem. This result implies that the problem can be solved in polynomial time by existing algorithms for the submodular resource allocation problem. We present specialized implementations of the existing algorithms and analyze their running time.