Akihisa Tamura

The rooted tree embedding problem into points in the plane

In this paper we show that any rooted tree ofn vertices can be straight-line embedded into any setS ofn points in the plane in general position so that the image of the root is arbitrarily specified.

A Two-Sided Discrete-Concave Market with Possibly Bounded Side Payments: An Approach by Discrete Convex Analysis

The marriage model due to Gale and Shapley [Gale, D., L. S. Shapley. 1962. College admissions and the stability of marriage. Amer. Math. Monthly69 9--15] and the assignment model due to Shapley and Shubik [Shapley, L. S., M. Shubik. 1972. The assignment game I: The core. Internat. J. Game Theory1 111--130] are standard in the theory of two-sided matching markets. We give a common generalization of these models by utilizing discrete-concave functions and considering possibly bounded side payments. We show the existence of a pairwise stable outcome in our model. Our present model is a further natural extension of the model examined in our previous paper [Fujishige, S., A. Tamura. A general two-sided matching market with discrete concave utility functions. Discrete Appl. Math.154 950--970], and the proof of the existence of a pairwise stable outcome is even simpler than the previous one.

A general two-sided matching market with discrete concave utility functions

In the theory of two-sided matching markets there are two standard models: (i) the marriage model due to Gale and Shapley and (ii) the assignment model due to Shapley and Shubik. Recently, Eriksson and Karlander introduced a hybrid model, which was further generalized by Sotomayor. In this paper, we propose a common generalization of these models by utilizing the framework of discrete convex analysis introduced by Murota, and verify the existence of a pairwise-stable outcome in our general model.

Coordinatewise domain scaling algorithm for M-convex function minimization

Abstract.We present a polynomial time scaling algorithm for the minimization of an M-convex function. M-convex functions are nonlinear discrete functions with (poly)matroid structures, which are being recognized as playing a fundamental role in tractable cases of discrete optimization. The algorithm is applicable also to a variant of quasi M-convex functions.

Combinatorial face enumeration in arrangements and oriented matroids

Abstract Let fk( F ) denote the number of k-dimensional faces of a d-dimensional arrangement F of spheres or a d-dimensional oriented matroid F . In this paper we show that the following relation among the face numbers is valid: fk( F )≤(dk)fd( F ) for 0≤k≤d. The same inequalities are valid for d-dimensional arrangements of hyperplanes. Using the result, we obtain a polynomial algorithm to enumerate all faces from the set of maximal faces of an oriented matroid. This algorithm can be applied to any arrangement of hyperplanes in projective space Pd or in Euclidean space Ed. Combining this with a recent result of Cordovil and Fukuda, we have the following: given the cograph of an arrangement (where the vertices are the d-faces and two vertices are adjacent if they intersect in a (d−1)-face), one can reconstruct the location vectors of all faces of the arrangement up to isomorphism in polynomial time. It is also shown that one can test in polynomial time whether a given set of (+,0,−)-vectors is the set of maximal vectors (topes) of an oriented matroid.

A Generalized Gale-Shapley Algorithm for a Discrete-Concave Stable-Marriage Model

The stable marriage model due to Gale and Shapley is one of the most fundamental two-sided matching models. Recently, Fleiner generalized the model in terms of matroids, and Eguchi and Fujishige extended the matroidal model to the framework of discrete convex analysis. In this paper, we extend their model to a vector version in which indifference on preferences is allowed, and show the existence of a stable solution by a generalization of the Gale-Shapley algorithm.

Designing matching mechanisms under constraints: An approach from discrete convex analysis

In this paper, we consider two-sided, many-to-one matching problems where agents in one side of the market (hospitals) impose some distributional constraints (e.g., a minimum quota for each hospital). We show that when the preference of the hospitals is represented as an M-natural-concave function, the following desirable properties hold: (i) the time complexity of the generalized GS mechanism is O(|X|^3), where |X| is the number of possible contracts, (ii) the generalized Gale & Shapley (GS) mechanism is strategyproof, (iii) the obtained matching is stable, and (iv) the obtained matching is optimal for the agents in the other side (doctors) within all stable matchings. Furthermore, we clarify sufficient conditions where the preference becomes an M-natural-concave function. These sufficient conditions are general enough so that they can cover most of existing works on strategyproof mechanisms that can handle distributional constraints in many-to-one matching problems. These conditions provide a recipe for non-experts in matching theory or discrete convex analysis to develop desirable mechanisms in such settings.

EP theorems and linear complementarity problems

Let A be a rational n × n square matrix and b be a rational n-vector for some positive integer n. The linear complementarity problem (abbreviated by LCP) is to find a vector (x, y) in R^(2n) satisfying y = Ax + b (x, y) >= 0 and the complementarity condition: xi yi = 0 for all i = 1 ,... , n. The LCP is known to be NP-complete, but there are some known classes of matrices A for which the LCP is polynomially solvable, for example the class of positive semi-definite (PSD-) matrices. In this paper, we study the LCP from the view point of EP (existentially polynomial time) theorems due to Cameron and Edmonds. In particular, we investigate the LCP duality theorem of Fukuda and Terlaky in EP form, and show that this immediately yields a simple modification of the criss- cross method with a nice practical feature. Namely, this algorithm can be applied to any given A and b, and terminates in one of the three states: (1) a solution x is found; (2) a solution to the dual LCP is found (implying the nonexistence of a solution to the LCP); or (3) a succinct certificate is given to show that the input matrix A is not “sufficient”. Note that all PSD-matrices and P- matrices are sufficient matrices.

Degree constrained tree embedding into points in the plane

Abstract Given a set N = {p1,…,pn} of n points in general position in the plane, and a positive integral n-vector d = (d 1 ,…,d n ) satisfying ∑ni=1di=2n − 2, can we construct a tree on N, such that the degree of point pi is di and none of the (n − 1) line segments connecting two points corresponding to endpoints of an edge intersect each other (except possibly at its endpoints)? We give a simple proof of the existence of such a tree in any instance and propose an algorithm polynomial on n for constructing one.

On the existence of sports schedules with multiple venues

We give theoretical methods of creating sports schedules where there are multiple venues for the games, and the number of times each team uses each venue should be balanced. A construction for leagues having 2^p>=8 teams was given by de Werra, Ekim and Raess. Here we show that feasible schedules exist when the league has an arbitrary even number of teams greater than or equal to 8.