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Akiyoshi Shioura

Publications and source records attributed to Akiyoshi Shioura.

8 recordsLinked to original sources
arXiv2026

Extension of Excess Demand Ascending Auction to Multi-Demand Model by Discrete Convex Analysis Approach

We consider the problem of finding the (unique) minimal Walrasian equilibrium price in multi-item, multi-unit auction models: there are multiple indivisible items for sale, with several units of each item, and a bidder may be interested in buying more than one copy of each item. In its special case with unit-demand bidders, where each bidder demands at most one unit of any item, Andersson, Andersson, and Talman (2013) proposed a general framework of ascending auction algorithms based on the concept of excess-demand item set. This paper extends this approach to the multi-unit case by exploiting the discrete convexity of the Lyapunov function associated with the auction model. In particular, we make use of the facts that (i) the equilibrium price vectors are characterized as the minimizers of the Lyapunov function, (ii) the Lyapunov function is an instance of an L-natural-convex function, and (iii) a concept generalizing ``excess-demand item set'' can be defined in L-natural-convex function minimization in general.

math.CO↗
arXiv2025

Steepest Descent Algorithm for M-convex Function Minimization Using Long Step Length

We consider the minimization of an M-convex function, which is a discrete convexity concept for functions on the integer lattice points. It is known that a minimizer of an Mconvex function can be obtained by the steepest descent algorithm. In this paper, we propose an effective use of long step length in the steepest descent algorithm, aiming at the reduction in the running time. In particular, we obtain an improved time bound by using long step length. We also consider the constrained M-convex function minimization and show that long step length can be applied to a variant of steepest descent algorithm as well.

math.OC↗
arXiv2024

An Improved Proximity Bound for Bike-Dock Reallocation Problem in Bike Sharing System

We consider a class of nonlinear integer programming problems arising from re-allocation of dock-capacity in a bike sharing system. The main aim of this note is to derive an improved proximity bound for the problem and its scaled variant. This makes it possible to refine the time bound for the polynomial-time proximity-scaling algorithm by Freund et al. (2022).

math.OC↗
arXiv2023

Note on Minimization of Quasi M$^\natural$-convex Functions

For a class of discrete quasi convex functions called semi-strictly quasi M$^\natural$-convex functions, we investigate fundamental issues relating to minimization, such as optimality condition by local optimality, minimizer cut property, geodesic property, and proximity property. Emphasis is put on comparisons with (usual) M$^\natural$-convex functions. The same optimality condition and a weaker form of the minimizer cut property hold for semi-strictly quasi M$^\natural$-convex functions, while geodesic property and proximity property fail.

math.CO↗
arXiv2023

Note on Steepest Descent Algorithm for Quasi L$^{\natural}$-convex Function Minimization

We define a class of discrete quasi convex functions, called semi-strictly quasi L$^{\natural}$-convex functions, and show that the steepest descent algorithm for L$^{\natural}$-convex function minimization also works for this class of quasi convex functions. The analysis of the exact number of iterations is also extended, revealing the so-called geodesic property of the steepest descent algorithm when applied to semi-strictly quasi L$^{\natural}$-convex functions.

math.OC↗
arXiv2018

M-convex Function Minimization Under L1-Distance Constraint

In this paper we consider a new problem of minimizing an M-convex function under L1-distance constraint (MML1); the constraint is given by an upper bound for L1-distance between a feasible solution and a given "center." This is motivated by a nonlinear integer programming problem for re-allocation of dock capacity in a bike sharing system discussed by Freund et al. (2017). The main aim of this paper is to better understand the combinatorial structure of the dock re-allocation problem through the connection with M-convexity, and show its polynomial-time solvability using this connection. For this, we first show that the dock re-allocation problem can be reformulated in the form of (MML1). We then present a pseudo-polynomial-time algorithm for (MML1) based on steepest descent approach. We also propose two polynomial-time algorithms for (MML1) by replacing the L1-distance constraint with a simple linear constraint. Finally, we apply the results for (MML1) to the dock re-allocation problem to obtain a pseudo-polynomial-time steepest descent algorithm and also polynomial-time algorithms for this problem. The proposed algorithm is based on a proximity-scaling algorithm for a relaxation of the dock re-allocation problem, which is of interest in its own right.

math.OC↗
arXiv2016

Colored Spanning Graphs for Set Visualization

We study an algorithmic problem that is motivated by ink minimization for sparse set visualizations. Our input is a set of points in the plane which are either blue, red, or purple. Blue points belong exclusively to the blue set, red points belong exclusively to the red set, and purple points belong to both sets. A \emph{red-blue-purple spanning graph} (RBP spanning graph) is a set of edges connecting the points such that the subgraph induced by the red and purple points is connected, and the subgraph induced by the blue and purple points is connected. We study the geometric properties of minimum RBP spanning graphs and the algorithmic problems associated with computing them. Specifically, we show that the general problem can be solved in polynomial time using matroid techniques. In addition, we discuss more efficient algorithms for the case in which points are located on a line or a circle, and also describe a fast $(\frac 12ρ+1)$-approximation algorithm, where $ρ$ is the Steiner ratio.

cs.CG↗