Inapproximability Results for Scheduling with Interval and Resource Restrictions

In the restricted assignment problem, the input consists of a set of machines and a set of jobs each with a processing time and a subset of eligible machines. The goal is to find an assignment of the jobs to the machines minimizing the makespan, that is, the maximum summed up processing time any machine receives. Herein, jobs should only be assigned to those machines on which they are eligible. It is well-known that there is no polynomial time approximation algorithm with an approximation guarantee of less than 1.5 for the restricted assignment problem unless P=NP. In this work, we show hardness results for variants of the restricted assignment problem with particular types of restrictions. For the case of interval restrictions - where the machines can be totally ordered such that jobs are eligible on consecutive machines - we show that there is no polynomial time approximation scheme (PTAS) unless P=NP. The question of whether a PTAS for this variant exists was stated as an open problem before, and PTAS results for special cases of this variant are known. Furthermore, we consider a variant with resource restriction where the sets of eligible machines are of the following form: There is a fixed number of (renewable) resources, each machine has a capacity, and each job a demand for each resource. A job is eligible on a machine if its demand is at most as big as the capacity of the machine for each resource. For one resource, this problem has been intensively studied under several different names and is known to admit a PTAS, and for two resources the variant with interval restrictions is contained as a special case. Moreover, the version with multiple resources is closely related to makespan minimization on parallel machines with a low rank processing time matrix. We show that there is no polynomial time approximation algorithm with a rate smaller than 48/47 ? 1.02 or 1.5 for scheduling with resource restrictions with 2 or 4 resources, respectively, unless P=NP. All our results can be extended to the so called Santa Claus variants of the problems where the goal is to maximize the minimal processing time any machine receives

Size-dependent nonlocal effects in plasmonic semiconductor particles

Localized surface plasmons (LSP) in semiconductor particles are expected to exhibit spatial nonlocal response effects as the geometry enters the nanometer scale. To investigate these nonlocal effects, we apply the hydrodynamic model to nanospheres of two different semiconductor materials: intrinsic InSb and $n$-doped GaAs. Our results show that the semiconductors indeed display nonlocal effects, and that these effects are even more pronounced than in metals. In a $150\mathrm{\,nm}$ InSb particle at $300\mathrm{\,K}$, the LSP frequency is blueshifted 35%, which is orders of magnitude larger than the blueshift in a metal particle of the same size. This property, together with their tunability, makes semiconductors a promising platform for experiments in nonlocal effects.Comment: 7 pages, 3 figures, 1 table, corrected typos in text and figure

Two-fluid hydrodynamic model for semiconductors

The hydrodynamic Drude model (HDM) has been successful in describing the optical properties of metallic nanostructures, but for semiconductors where several different kinds of charge carriers are present, an extended theory is required. We present a two-fluid hydrodynamic model for semiconductors containing electrons and holes (from thermal or external excitation) or light and heavy holes (in $p$-doped materials). The two-fluid model predicts the existence of two longitudinal modes, an acoustic and an optical, whereas only an optical mode is present in the HDM. By extending nonlocal Mie theory to two plasmas, we are able to simulate the optical properties of two-fluid nanospheres and predict that the acoustic mode gives rise to peaks in the extinction spectra that are absent in the HDM.Comment: Accepted in PRB. 17 pages, 9 figures, 1 tabl

A Group Contingency Plus Self-Management Intervention Targeting At-Risk Secondary Students' Class-work and Active Responding

Incomplete written work and lack of active classroom responding are reported to be obstacles to secondary students' learning. Effective interventions found in meta-analytic reviews of the current research literature include: differential reinforcement of desired behaviors through group contingencies and self-management strategies (Hoagwood et al., 2007; Prout & Prout, 198; Stage & Quiroz, 1997). However, the current studies have mostly focused on elementary school settings. The purpose of the present study is to show that an independent group contingency combined with self-management strategies and randomized-reinforcer components can increase the amount of written work and active classroom responding in high school students. Three remedial reading classes with a total of 15 students participated in this study. Students used self-management strategies during independent reading time to increase the amount of writing in their reading logs. They used self-monitoring strategies to record whether or not they performed expected behaviors in class. A token economy was used to provide positive reinforcement for target responses. The results were analyzed through visual inspection of graphs and effect size computations and showed that the intervention increased the total amount of written words in the students' reading logs and overall classroom academic responding

Approximation Schemes for Machine Scheduling

In the classical problem of makespan minimization on identical parallel machines, or machine scheduling for short, a set of jobs has to be assigned to a set of machines. The jobs have a processing time and the goal is to minimize the latest finishing time of the jobs. Machine scheduling is well known to be NP-hard and thus there is no polynomial time algorithm for this problem that is guaranteed to find an optimal solution unless P=NP. There is, however, a polynomial time approximation scheme (PTAS) for machine scheduling, that is, a family of approximation algorithms with ratios arbitrarily close to one. Whether a problem admits an approximation scheme or not is a fundamental question in approximation theory. In the present work, we consider this question for several variants of machine scheduling. We study the problem where the machines are partitioned into a constant number of types and the processing time of the jobs is also dependent on the machine type. We present so called efficient PTAS (EPTAS) results for this problem and variants thereof. We show that certain cases of machine scheduling with assignment restrictions do not admit a PTAS unless P=NP. Moreover, we introduce a graph framework based on the restrictions of the jobs and use it in the design of approximation schemes for other variants. We introduce an enhanced integer programming formulation for assignment problems, show that it can be efficiently solved, and use it in the EPTAS design for variants of machine scheduling with setup times. For one of the problems, we show that there is also a PTAS in the case with uniform machines, where machines have speeds influencing the processing times of the jobs. We consider cases in which each job requires a certain amount of a shared renewable resource and the processing time is depended on the amount of resource it receives or not. We present so called asymptotic fully polynomial time approximation schemes (AFPTAS) for the problems

Reduced Models of Point Vortex Systems in Quasigeostrophic Fluid Dynamics

We develop a nonequilibrium statistical mechanical description of the evolution of point vortex systems governed by either the Euler, single-layer quasigeostrophic or two-layer quasigeostrophic equations. Our approach is based on a recently proposed optimal closure procedure for deriving reduced models of Hamiltonian systems. In this theory the statistical evolution is kept within a parametric family of distributions based on the resolved variables chosen to describe the macrostate of the system. The approximate evolution is matched as closely as possible to the true evolution by minimizing the mean-squared residual in the Liouville equation, a metric which quantifies the information loss rate due to model reduction. The point vortex approximation of the fluid dynamics allows the optimal closure, which is formulated on phase space, to be transferred to physical space resulting in an exact mean-field theory for the continuum limit. The near-equilibrium linearization of this theory is used to model the intrinsic relaxation rates for isolated coherent vorticity structures. The equilibration of initially disturbed vorticity fields is captured by a reduced model that has few resolved variables and no adjustable parameters. For the Euler and single-layer quasigeostrophic equations, the theory is used to model the axisymmetrization of a deformed vorticity patch. In particular, the reduced model exhibits how the rate of symmetrization depends upon the energy and the Rossby deformation radius. For the two-layer equations the study focuses on the relaxation of baroclinic perturbations of stable barotropic structures and the transfer of available potential energy to kinetic energy. The model predicts the dependence of the barotropization rate on the energy and the internal Rossby deformation radius. Both axisymmetrization and barotropization are prominent features of the coherent vortex structures observed in direct numerical simulations of two-dimensional and quasigeostrophic turbulence. The reduced model is tested against the evolution of an ensemble of point vortex systems to validate its predictions. Therefore, the reduced model furnishes a mathematical theory of these fluid dynamical phenomena

Online Load Balancing on Uniform Machines with Limited Migration

In the problem of online load balancing on uniformly related machines with bounded migration, jobs arrive online one after another and have to be immediately placed on one of a given set of machines without knowledge about jobs that may arrive later on. Each job has a size and each machine has a speed, and the load due to a job assigned to a machine is obtained by dividing the first value by the second. The goal is to minimize the maximum overall load any machine receives. However, unlike in the pure online case, each time a new job arrives it contributes a migration potential equal to the product of its size and a certain migration factor. This potential can be spend to reassign jobs either right away (non-amortized case) or at any later time (amortized case). Semi-online models of this flavor have been studied intensively for several fundamental problems, e.g., load balancing on identical machines and bin packing, but uniformly related machines have not been considered up to now. In the present paper, the classical doubling strategy on uniformly related machines is combined with migration to achieve an $(8/3+\varepsilon)$-competitive algorithm and a $(4+\varepsilon)$-competitive algorithm with $O(1/\varepsilon)$ amortized and non-amortized migration, respectively, while the best known competitive ratio in the pure online setting is roughly $5.828$

Hydrodynamic acoustic plasmon resonances in semiconductor nanowires and their dimers

The hydrodynamic Drude model known from metal plasmonics also applies to semiconductor structures of sizes in between single-particle quantum confinement and bulk. But contrary to metals, for semiconductors two or more types of plasma may have to be taken into account in order to properly describe their plasmonic properties. In this combined analytical and computational study, we explore predictions of the recently proposed two-fluid hydrodynamic Drude model for the optical properties of plasmonic semiconductor nanowires, in particular for thermally excited InSb nanowires. We focus on the low-frequency acoustic surface and bulk plasmon resonances that are unique fingerprints for this model and are yet to be observed. We identify these resonances in spectra for single nanowires based on analytical calculations, and they are in complete agreement with our numerical implementation of the model. For dimers of nanowires we predict substantial increase of the extinction cross section and field enhancement of the acoustic localized surface plasmon resonance, which makes its observation in dimers more likely.Comment: I would like to inform that Dr.Abbas Zarifi is the corresponding author of this pape