Satoru Fujishige

A combinatorial strongly polynomial algorithm for minimizing submodular functions

This paper presents a combinatorial polynomial-time algorithm for minimizing submodular functions, answering an open question posed in 1981 by Grötschel, Lovász, and Schrijver. The algorithm employs a scaling scheme that uses a flow in the complete directed graph on the underlying set with each arc capacity equal to the scaled parameter. The resulting algorithm runs in time bounded by a polynomial in the size of the underlying set and the length of the largest absolute function value. The paper also presents a strongly polynomial version in which the number of steps is bounded by a polynomial in the size of the underlying set, independent of the function values.

Polymatroidal Dependence Structure of a Set of Random Variables

Given a finite set E of random variables, the entropy function h on E is a mapping from the set of all subsets of E into the set of all nonnegative real numbers such that for each A ⊆ E h(A) is the entropy of A . The present paper points out that the entropy function h is a β -function, i.e., a monotone non-decreasing and submodular function with h(O) = 0 and that the pair ( E, h ) is a polymatroid. The polymatroidal structure of a set of random variables induced by the entropy function is fundamental when we deal with the interdependence analysis of random variables such as the information-theoretic correlative analysis, the analysis of multiple-user communication networks, etc. Also, we introduce the notion of the principal partition of a set of random variables by transferring some results in the theory of matroids.

Lexicographically Optimal Base of a Polymatroid with Respect to a Weight Vector

Let ( E , (rho)) be a polymatroid with a ground set E and a rank function (rho). A base x = ( x ( e )) (epsilon) (in) E of polymatroid ( E , (rho)) is called a lexicographically optimal base of ( E , (rho)) with respect to a weight vector w = ( w ( e )) (epsilon)(in) E if the | E |-tuple of the numbers x ( e )/ w ( e )( e (in) E ) arranged in order of increasing magnitude is lexicographically maximum among all | E |-tuples of numbers y ( e )/ w ( e )( e (in) E ) arranged in the same manner for all bases y = ( y ( e )) e (in) E of ( E , (rho)). We give theorems that characterize the relationship between weight vectors and lexicographically optimal bases and point out that a lexicographically optimal base minimizes among all bases a quadratic objective function defined in terms of the associated weight vector. Also, we present an algorithm for finding the (unique) lexicographically optimal base with respect to a given weight vector. Furthermore, we consider the problem of determining the set of weight vectors with respect to which a given base is lexicographically optimal and provide an algorithm for solving it, which is useful for the sensitivity analysis of the optimal base with regard to the variation of the weight vector. The algorithms proposed in the present paper efficiently solve the problem, treated by N . Megiddo, of finding a lexicographically optimal flow in a network with multiple sources and sinks, which is a special case of the problem considered here.

A Note on Kelso and Crawford's Gross Substitutes Condition

In their 1982 article, Kelso and Crawford proposed a gross substitutes condition for the existence of core (and equilibrium) in a two-sided matching model. Since then, this condition has often been used in the literature on matching models and equilibrium models in the presence of indivisibilities. In this paper we prove that a reservation value (or utility) function satisfies the gross substitutes condition if and only if it is anM?-concave function defined on the unit-hypercube, which is a discrete concave function recently introduced by Murota and Shioura (1999).

Notes on L-/M-convex functions and the separation theorems

Abstract.The concepts of L-convex function and M-convex function have recently been introduced by Murota as generalizations of submodular function and base polyhedron, respectively, and discrete separation theorems are established for L-convex/concave functions and for M-convex/concave functions as generalizations of Frank’s discrete separation theorem for submodular/supermodular set functions and Edmonds’ matroid intersection theorem. This paper shows the equivalence between Murota’s L-convex functions and Favati and Tardella’s submodular integrally convex functions, and also gives alternative proofs of the separation theorems that provide a geometric insight by relating them to the ordinary separation theorem in convex analysis.

Sequential state estimation with interrupted observation

This paper deals with the state estimation problem for linear discrete time systems with the interrupted observation mechanisms which can be characterized by the independent binary sequence taking on the values of 0 or 1. On the basis of the Bayesian approach, the approximate minimum variance estimator is derived for the case when at any time the probability of occurrence of interruption is known a priori. An adaptive estimator algorithm is also established when the probability of interruption is unknown but fixed throughout the time interval of interest. Unlike the usual Kalman filter algorithm, all the estimators derived here are nonlinear with respect to observations and the associated covariance equations are related to the actual observations. Digital simulation studies are demonstrated to compare the performance of the approximate nonlinear estimator presented here with that of the best linear estimator due to Nahi (1969) .

A combinatorial, strongly polynomial-time algorithm for minimizing submodular functions

This paper presents a combinatorial polynomial-time algorithm for minimizing submodular functions, answering an open question posed in 1981 by Grotschel, Lov asz, and Schrijver. The algorithm employs a scaling scheme that uses a ow in the complete directed graph on the underlying set with each arc capacity equal to the scaled parameter. The resulting algorithm runs in time bounded by a polyno- mial in the size of the underlying set and the length of the largest absolute function value. The paper also presents a strongly polynomial version in which the number of steps is bounded by a polynomial in the size of the underlying set, independent of the function values.

A capacity-rounding algorithm for the minimum-cost circulation problem: A dual framework of the Tardos algorithm

Recently, É. Tardos gave a strongly polynomial algorithm for the minimum-cost circulation problem and solved the open problem posed in 1972 by J. Edmonds and R.M. Karp. Her algorithm runs in O(m2T(m, n) logm) time, wherem is the number of arcs,n is the number of vertices, andT(m, n) is the time required for solving a maximum flow problem in a network withm arcs andn vertices. In the present paper, taking an approach that is a dual of Tardoss, we also give a strongly polynomial algorithm for the minimum-cost circulation problem. Our algorithm runs in O(m2S(m, n) logm) time and reduces the computational complexity, whereS(m, n) is the time required for solving a shortest path problem with a fixed origin in a network withm arcs,n vertices, and a nonnegative arc length function. The complexity is the same as that of Orlins algorithm, recently developed by efficiently implementing the Edmonds-Karp scaling algorithm.

Theory of submodular programs: A fenchel-type min-max theorem and subgradients of submodular functions

We consider submodular programs which are problems of minimizing submodular functions on distributive lattices with or without constraints. We define a convex (or concave) conjugate function of a submodular (or supermodular) function and show a Fenchel-type min-max theorem for submodular and supermodular functions. We also define a subgradient of a submodular function and derive a necessary and sufficient condition for a feasible solution of a submodular program to be optimal, which is a counterpart of the Karush-Kuhn-Tucker condition for convex programs.

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.