Akiyoshi Shioura

M-Convex Function on Generalized Polymatroid

The concept of M-convex function, introduced by Murota 1996, 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 and 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.

Relationship of M-/L-convex functions with discrete convex functions by Miller and Favati-Tardella

Abstract We clarify the relationship of the concepts of M-convex and L-convex functions due to Murota (Adv. Math. 124 (1996); Math. Programming 83 (1998)) with two other concepts of discrete convex functions over integer lattice points, discretely-convex functions due to Miller (SIAM J. Appl. Math. 21 (1971)), and integrally-convex functions due to Favati–Tardella (Ricerca Operativa 53 (1990)). We also investigate whether each class of discrete convex functions is closed under fundamental operations such as addition and convolution.

Minimization of an M-convex function

We study the minimization of an M-convex function introduced by Murota. It is shown that any vector in the domain can be easily separated from a minimizer of the function. Based on this property, we develop a polynomial time algorithm.

Conjugacy relationship between M-convex and L-convex functions in continuous variables

Abstract.By extracting combinatorial structures in well-solved nonlinear combinatorial optimization problems, Murota (1996,1998) introduced the concepts of M-convexity and L-convexity to functions defined over the integer lattice. Recently, Murota–Shioura (2000, 2001) extended these concepts to polyhedral convex functions and quadratic functions in continuous variables. In this paper, we consider a further extension to more general convex functions defined over the real space, and provide a proof for the conjugacy relationship between general M-convex and L-convex functions.

Quasi M-convex and L-convex functions: quasiconvexity in discrete optimization

We introduce two classes of discrete quasiconvex functions, called quasi M- and L-convex functions, by generalizing the concepts of M- and L-convexity due to Murota (Adv. Math. 124 (1996) 272) and (Math. Programming 83 (1998) 313). We investigate the structure of quasi Mand L-convex functions with respect to level sets, and show that various greedy algorithms work for the minimization of quasi M- and L-convex functions.

ON THE PIPAGE ROUNDING ALGORITHM FOR SUBMODULAR FUNCTION MAXIMIZATION — A VIEW FROM DISCRETE CONVEX ANALYSIS

We consider the problem of maximizing a nondecreasing submodular set function under a matroid constraint. Recently, Calinescu et al. (2007) proposed an elegant framework for the approximation of this problem, which is based on the pipage rounding technique by Ageev and Sviridenko (2004), and showed that this framework indeed yields a (1 - 1/e)-approximation algorithm for the class of submodular functions which are represented as the sum of weighted rank functions of matroids. This paper sheds a new light on this result from the viewpoint of discrete convex analysis by extending it to the class of submodular functions which are the sum of M♮-concave functions. M♮-concave functions are a class of discrete concave functions introduced by Murota and Shioura (1999), and contain the class of the sum of weighted rank functions as a proper subclass. Our result provides a better understanding for why the pipage rounding algorithm works for the sum of weighted rank functions. Based on the new observation, we further extend the approximation algorithm to the maximization of a nondecreasing submodular function over an integral polymatroid. This extension has an application in multi-unit combinatorial auctions.

Polynomial-Time Algorithms for Linear and Convex Optimization on Jump Systems

The concept of jump system, introduced by Buchet and Cunningham (1995), is a set of integer points with a certain exchange property. In this paper, we discuss several linear and convex optimization problems on jump systems and show that these problems can be solved in polynomial time under the assumption that a membership oracle for a jump system is available. We firstly present a polynomial-time implementation of the greedy algorithm for the minimization of a linear function. We then consider the minimization of a separable-convex function on a jump system, and propose the first polynomial-time algorithm for this problem. The algorithm is based on the domain reduction approach developed in Shioura (1998). We finally consider the concept of M-convex functions on constant-parity jump systems which has been recently proposed by Murota (2006). It is shown that the minimization of an M-convex function can be solved in polynomial time by the domain reduction approach.

Exact bounds for steepest descent algorithms of L-convex function minimization

We analyze minimization algorithms for L^@?-convex functions in discrete convex analysis and establish exact bounds for the number of iterations required by the steepest descent algorithm and its variants.

The tree center problems and the relationship with the bottleneck knapsack problems

The tree center problems are designed to find a subtree minimizing the maximum distance from any vertex. This paper shows that these problems in a tree network are related to the bottleneck knapsack problems and presents linear-time algorithms for the tree center problems by using the relation.

A Linear Time Algorithm for Finding ak-Tree Core

Given a tree containingnvertices, consider the sum of the distance between all vertices and ak-leaf subtree (subtree which contains exactlykleaves). Ak-tree core is ak-leaf subtree which minimizes the sum of the distances. In this paper, we propose a linear time algorithm for finding ak-tree core for a givenk.