Kazutoshi Ando

On structures of bisubmodular polyhedra

A bisubmodular polyhedron is defined in terms of a so-called bisubmodular function on a family of ordered pairs of disjoint subsets of a finite set. We examine the structures of bisubmodular polyhedra in terms of signed poset and exchangeability graph. We give a characterization of extreme points together with an O(n2) algorithm for discerning whether a given point is an extreme point, wheren is the cardinality of the underlying set, and we assume a function evaluation oracle for the bisubmodular function. The algorithm also determines the signed posetructure associated with the given point if it is an extreme point. We reveal the adjacency relation of extreme points by means of the Hasse diagrams of the associated signed posets. Moreover, we investigate the connectivity and the decomposition of a bisubmodular system into its connected components.

A characterization of bisubmodular functions

Abstract Given a nonempty finite set E , let 3 E be the set of all the ordered pairs of disjoint subsets of E . We call a function on 3 E a biset function. A biset function f:3 E → R is called bisubmodular if we have ∀(X 1 ,Y 1 ), (X 2 ,Y 2 ) ϵ 3 E : ƒ(X 1 ,Y 1 ) + ƒ(X 2 ,Y 2 )⩾ƒ(X 1 ∪ Y 2 ) − (Y 1 ∪ Y 2 ),(Y 1 ∪ Y 2 ) − (X 1 ∪ x 2 )) +ƒ(X 1 ∩ X 2 , Y 1 ∩ Y 2 ). We give a simple necessary and sufficient condition for a biset function to be bisubmodular.

Decomposition of a bidirected graph into strongly connected components and its signed poset structure

Abstract A bidirected graph is a graph each arc of which has either two positive end-vertices (tails), two negative end-vertices (heads), or one positive end-vertex (a tail) and one negative end-vertex (a head). We define the strong connectivity of a bidirected graph as a generalization of the strong connectivity of an ordinary directed graph. We show that a bidirected graph is decomposed into strongly connected components and that a signed poset structure is naturally defined on the set of the consistent strongly connected components. We also give a linear time algorithm for decomposing a bidirected graph into strongly connected components. Furthermore, we discuss the relationship between the decomposition of a bidirected graph and the minimization of a certain bisubmodular function.

Strategy-proof and symmetric allocation of an indivisible good

Abstract We consider economies with a single indivisible good and money. We study the possibility of constructing strategy-proof, symmetric, and budget balanced mechanisms. We show three impossibility results on restricted domains: there is no strategy-proof, symmetric, and budget balanced mechanism satisfying either (i) equal compensation, (ii) normal compensation, or (iii) individual rationality. Moreover, the first result is strengthened by replacing symmetry with weak symmetry. In contrast, we show that our sequential mechanisms satisfy strategy-proofness, weak symmetry, budget balance, and the other three axioms on much more restricted domains.

K-submodular functions and convexity of their Lovász extension

We consider a class of lattice polyhedra introduced by Hoffman and Schwartz. The polyhedra are defined in terms of a kind of submodular function defined on the set of antichains of a poset. Recently, Kruger (Discrete Appl. Math. 99 (2000) 125-148) showed the validity of a greedy algorithm for this class of lattice polyhedra, which had been proved by Faigle and Kern to be valid for a less general class of polyhedra. In this paper, we investigate submodular functions in Krugers sense and associated polyhedra. We show that the Lovasz extension of a submodular function in Krugers sense is convex, and vice versa. Furthermore, we show a polynomial-time algorithm to test whether or not a vector is an extreme point of the associated polyhedron.

Monotonicity of minimum distance inefficiency measures for Data Envelopment Analysis

This research explores the minimum distance inefficiency measure for the Data Envelopment Analysis (DEA) model. A critical issue is that this measure does not satisfy monotonicity, i.e., the measure may provide a better evaluation score to an inferior decision making unit (DMU) than to a superior one. To overcome this, a variant called the extended facet approach has been introduced. This approach, however, requires a certain regularity condition to be met. We discuss several special classes of the DEA model, and show that for these models, the minimum distance inefficiency measure satisfies the monotonicity property without the regularity condition. Moreover, we conducted computational experiments using real-world data sets from these special classes, and demonstrated that the extended facet approach may overestimate the performance of a DMU.

An algorithm for finding a representation of a subtree distance

Generalizing the concept of tree metric, Hirai (2006) introduced the concept of subtree distance. A mapping \(d:X\times X \rightarrow \mathbb {R}_+\) is called a subtree distance if there exist a weighted tree T and a family \(\{T_x\,|\,x \in X\}\) of subtrees of T indexed by the elements in X such that \(d(x,y)=d_T(T_x,T_y)\), where \(d_T(T_x,T_y)\) is the distance between \(T_x\) and \(T_y\) in T. Hirai (2006) gave a characterization of subtree distances which corresponds to Buneman’s four-point condition (1974) for the tree metrics. Using this characterization, we can decide whether or not a given matrix is a subtree distance in O\((n^4)\) time. However, the existence of a polynomial time algorithm for finding a tree and subtrees representing a subtree distance has been an open question. In this paper, we show an O\((n^3)\) time algorithm that finds a tree and subtrees representing a given subtree distance.

Efficient algorithms for subdominant cycle-complete cost functions and cycle-complete solutions

Abstract The cycle-complete solution introduced by Trudeau (2012) is a solution concept for minimum cost spanning tree games and was proved to have desirable properties such as core-selection and sensitivity to change of the cost function. The cycle-complete solution is defined as the Shapley value of the minimum cost spanning tree game associated with the subdominant cycle-complete cost function of a given cost function. In this study, we characterize subdominant cycle-complete cost functions and provide an O ( n 2 log n ) time algorithm for computing such functions, where n is the number of players. This algorithm leads to a new algorithm for computing the cycle-complete solution of a minimum cost spanning tree game with an O ( n 2 log n ) time bound.