Finding the Minimum-Weight k-Path

Given a weighted $n$-vertex graph $G$ with integer edge-weights taken from a range $[-M,M]$, we show that the minimum-weight simple path visiting $k$ vertices can be found in time \tilde{O}(2^k \poly(k) M n^\omega) = O^*(2^k M). If the weights are reals in $[1,M]$, we provide a $(1+\varepsilon)$-approximation which has a running time of \tilde{O}(2^k \poly(k) n^\omega(\log\log M + 1/\varepsilon)). For the more general problem of $k$-tree, in which we wish to find a minimum-weight copy of a $k$-node tree $T$ in a given weighted graph $G$, under the same restrictions on edge weights respectively, we give an exact solution of running time \tilde{O}(2^k \poly(k) M n^3) and a $(1+\varepsilon)$-approximate solution of running time \tilde{O}(2^k \poly(k) n^3(\log\log M + 1/\varepsilon)). All of the above algorithms are randomized with a polynomially-small error probability.Comment: To appear at WADS 201

50 Years of Test (Un)fairness: Lessons for Machine Learning

Quantitative definitions of what is unfair and what is fair have been introduced in multiple disciplines for well over 50 years, including in education, hiring, and machine learning. We trace how the notion of fairness has been defined within the testing communities of education and hiring over the past half century, exploring the cultural and social context in which different fairness definitions have emerged. In some cases, earlier definitions of fairness are similar or identical to definitions of fairness in current machine learning research, and foreshadow current formal work. In other cases, insights into what fairness means and how to measure it have largely gone overlooked. We compare past and current notions of fairness along several dimensions, including the fairness criteria, the focus of the criteria (e.g., a test, a model, or its use), the relationship of fairness to individuals, groups, and subgroups, and the mathematical method for measuring fairness (e.g., classification, regression). This work points the way towards future research and measurement of (un)fairness that builds from our modern understanding of fairness while incorporating insights from the past.Comment: FAT* '19: Conference on Fairness, Accountability, and Transparency (FAT* '19), January 29--31, 2019, Atlanta, GA, US

The Impatient May Use Limited Optimism to Minimize Regret

Discounted-sum games provide a formal model for the study of reinforcement learning, where the agent is enticed to get rewards early since later rewards are discounted. When the agent interacts with the environment, she may regret her actions, realizing that a previous choice was suboptimal given the behavior of the environment. The main contribution of this paper is a PSPACE algorithm for computing the minimum possible regret of a given game. To this end, several results of independent interest are shown. (1) We identify a class of regret-minimizing and admissible strategies that first assume that the environment is collaborating, then assume it is adversarial---the precise timing of the switch is key here. (2) Disregarding the computational cost of numerical analysis, we provide an NP algorithm that checks that the regret entailed by a given time-switching strategy exceeds a given value. (3) We show that determining whether a strategy minimizes regret is decidable in PSPACE

Boundary effects on one-particle spectra of Luttinger liquids

We calculate one-particle spectra for a variety of models of Luttinger liquids with open boundary conditions. For the repulsive Hubbard model the spectral weight close to the boundary is enhanced in a large energy range around the chemical potential. A power law suppression, previously predicted by bosonization, only occurs after a crossover at energies very close to the chemical potential. Our comparison with exact spectra shows that the effects of boundaries can partly be understood within the Hartree-Fock approximation.Comment: 4 pages including 4 figures, revised version, to be published in Phys. Rev. B, January 200

Об анатомическом строении членистостебельного растения Annulina Neuburgiana Radczenko

The purpose was to compare two approaches for the acquisition and analysis of dynamic-contrast-enhanced MRI data with respect to differences in the modelling of the arterial input-function (AIF), the dependency of the model parameters on physiological parameters and their numerical stability. Eight hundred tissue concentration curves were simulated for different combinations of perfusion, permeability, interstitial volume and plasma volume based on two measured AIFs and analysed according to the two commonly used approaches. The transfer constants (Approach 1) K (trans) and (Approach 2) k (ep) were correlated with all tissue parameters. K (trans) showed a stronger dependency on perfusion, and k (ep) on permeability. The volume parameters (Approach 1) v (e) and (Approach 2) A were mainly influenced by the interstitial and plasma volume. Both approaches allow only rough characterisation of tissue microcirculation and microvasculature. Approach 2 seems to be somewhat more robust than 1, mainly due to the different methods of CA administration

How universal is the one-particle Green's function of a Luttinger liquid?

The one-particle Green's function of the Tomonaga-Luttinger model for one-dimensional interacting Fermions is discussed. Far away from the origin of the plane of space-time coordinates the function falls off like a power law. The exponent depends on the direction within the plane. For a certain form of the interaction potential or within an approximated cut-off procedure the different exponents only depend on the strength of the interaction at zero momentum and can be expressed in terms of the Luttinger liquid parameters $K_{\rho}$ and $K_{\sigma}$ of the model at hand. For a more general interaction and directions which are determined by the charge velocity $v_{\rho}$ and spin velocity $v_{\sigma}$ the exponents also depend on the smoothness of the interaction at zero momentum and the asymptotic behavior of the Green's function is not given by the Luttinger liquid parameters alone. This shows that the physics of large space-time distances in Luttinger liquids is less universal than is widely believed.Comment: 5 pages with 2 figure