On the job rotation problem

The job rotation problem (JRP) is the following: Given an $n \times n$ matrix $A$ over \Re \cup \{\ -\infty\ \}\ and $k \leq n$, find a $k \times k$ principal submatrix of $A$ whose optimal assignment problem value is maximum. No polynomial algorithm is known for solving this problem if $k$ is an input variable. We analyse JRP and present polynomial solution methods for a number of special cases

Perturbation of eigenvalues of matrix pencils and optimal assignment problem

We consider a matrix pencil whose coefficients depend on a positive parameter $\epsilon$, and have asymptotic equivalents of the form $a\epsilon^A$ when $\epsilon$ goes to zero, where the leading coefficient $a$ is complex, and the leading exponent $A$ is real. We show that the asymptotic equivalent of every eigenvalue of the pencil can be determined generically from the asymptotic equivalents of the coefficients of the pencil. The generic leading exponents of the eigenvalues are the "eigenvalues" of a min-plus matrix pencil. The leading coefficients of the eigenvalues are the eigenvalues of auxiliary matrix pencils, constructed from certain optimal assignment problems.Comment: 8 page

Tropical bounds for eigenvalues of matrices

We show that for all k = 1,...,n the absolute value of the product of the k largest eigenvalues of an n-by-n matrix A is bounded from above by the product of the k largest tropical eigenvalues of the matrix |A| (entrywise absolute value), up to a combinatorial constant depending only on k and on the pattern of the matrix. This generalizes an inequality by Friedland (1986), corresponding to the special case k = 1.Comment: 17 pages, 1 figur

マックスプラス代数のスケジューリング問題への応用

学位の種別: 課程博士審査委員会委員 : （主査）東京大学教授 西成 活裕, 東京大学教授 太田 順, 東京大学教授 時弘 哲治, 東京大学准教授 白石 潤一, 東京大学准教授 柳澤 大地University of Tokyo(東京大学

Painleve equations and applications

The theme running throughout this thesis is the Painlevé equations, in their differential, discrete and ultra-discrete versions. The differential Painlevé equations have the Painlevé property. If all solutions of a differential equation are meromorphic functions then it necessarily has the Painlevé property. Any ODE with the Painlevé property is necessarily a reduction of an integrable PDE. Nevanlinna theory studies the value distribution and characterizes the growth of meromorphic functions, by using certain averaged properties on a disc of variable radius. We shall be interested in its well-known use as a tool for detecting integrability in difference equations—a difference equation may be integrable if it has sufficiently many finite-order solutions in the sense of Nevanlinna theory. This does not provide a sufficient test for integrability; additionally it must satisfy the well-known singularity confinement test. [Continues.

Protecting Micro-Data Privacy: The Moment-Based Density Estimation Method and its Application

Privacy concerns pertaining to the release of confidential micro-level information are increasingly relevant to organisations and institutions. Controlling the dissemination of disclosure-prone micro-data by means of suppression, aggregation and perturbation techniques often entails different levels of effectiveness and drawbacks depending on the context and properties of the data. In this dissertation, we briefly review existing disclosure control methods for microdata and undertake a study demonstrating the applicability of micro-data methods to proportion data. This is achieved by using the sample size efficiency related to a simple hypothesis test for a fixed significance level and power, as a measure of statistical utility. We compare a query-based differential privacy mechanism to the multiplicative noise method for disclosure control and demonstrate that with the correct specification of noise parameters, the multiplicative noise method, which is a micro-data based method, achieves similar disclosure protection properties with reduced statistical efficiency costs