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Noam Goldberg

Publications and source records attributed to Noam Goldberg.

10 recordsLinked to original sources
arXiv2026

A Parametrized Complexity View on Robust Scheduling with Budgeted Uncertainty

In this study, we investigate a robust single-machine scheduling problem under processing time uncertainty. The uncertainty is modeled using the budgeted approach, where each job has a nominal and deviation processing time, and the number of deviations is bounded by Γ. The objective is to minimize the number of tardy jobs where a job is considered tardy if there is some scenario in which it is completed after its due date. Since the problem is NP-hard in general, we focus on analyzing its tractability under the assumption that certain natural parameters of the problem are each bounded by a constant. We consider three parameters: the robustness parameter Γ, the number of distinct due dates in the instance, and the number of jobs with nonzero deviations. Using parameterized-complexity theory, we prove that the problem is W[1]-hard with respect to Γ, but can be solved in XP time with respect to the same parameter. With respect to the number of distinct due dates, we establish a stronger hardness result by showing that the problem remains NP-hard even when there are only two different due dates and is solvable in pseudo-polynomial time when the number of due dates is upper bounded by a constant. To complement these results, we show that the case of a common (single) due date reduces to a robust binary knapsack problem with equal item profits, a problem we prove to be solvable in polynomial time. Finally, we prove that the problem is fixed-parameter tractable with respect to the number of jobs with nonzero deviations.

cs.DM↗
arXiv2026

Robust Permutation Flowshops Under Budgeted Uncertainty

We consider the robust permutation flowshop problem under the budgeted uncertainty model, where at most a given number of job processing times may deviate on each machine. We show that solutions for this problem can be determined by solving polynomially many instances of the corresponding nominal problem. As a direct consequence, our result implies that this robust flowshop problem can be solved in polynomial time for two machines, and can be approximated in polynomial time for any fixed number of machines. The reduction that is our main result follows from an analysis similar to Bertsimas and Sim (2003) except that dualization is applied to the terms of a min-max objective rather than to a linear objective function. Our result may be surprising considering that heuristic and exact integer programming based methods have been developed in the literature for solving the two-machine flowshop problem. Next, we show a logarithmic factor improvement in the overall running time implied by a naive reduction to nominal problems in the case of two machines and three machines. We conclude by noting that our reduction appears to have more general consequences for robust optimization problems under budgeted uncertainty having a similar form.

cs.DS↗
arXiv2025

Robust Extensible Bin Packing and Revisiting the Convex Knapsack Problem

We study a robust extensible bin packing problem with budgeted uncertainty, under a budgeted uncertainty model where item sizes are defined to lie in the intersection of a box with a one-norm ball. We propose a scenario generation algorithm for this problem, which alternates between solving a master robust bin-packing problem with a finite uncertainty set and solving a separation problem. We first show that the separation is strongly NP-hard given solutions to the continuous relaxation of the master problem. Then, focusing on the separation problem for the integer master problem, we show that this problem becomes a special case of the continuous convex knapsack problem, which is known to be weakly NP-hard. Next, we prove that our special case when each of the functions is piecewise linear, having only two pieces, remains NP-hard. We develop a pseudo-polynomial dynamic program (DP) and a fully polynomial-time approximation scheme (FPTAS) for our special case whose running times match those of a binary knapsack FPTAS. Finally, our computational study shows that the DP can be significantly more efficient in practice compared with solving the problem with specially ordered set (SOS) constraints using advanced mixed-integer (MIP) solvers. Our experiments also demonstrate the application of our separation problem method to solving the robust extensible bin packing problem, including the evaluation of deferring the exact solution of the master problem, separating based on approximate master solutions in intermediate iterations. Finally, a case-study, based on real elective surgery data, demonstrates the potential advantage of our model compared with the actual schedule and optimal nominal schedules.

cs.DM↗
arXiv2025

Smooth Uncertainty Sets: Dependence of Uncertain Parameters via a Simple Polyhedral Set

We propose a novel polyhedral uncertainty set for robust optimization, termed the smooth uncertainty set, which captures dependencies of uncertain parameters by constraining their pairwise differences. The bounds on these differences may be dictated by the underlying physics of the problem and may be expressed by domain experts. When correlations are available, the bounds can be set to ensure that the associated probabilistic constraints are satisfied for any given probability. We explore specialized solution methods for the resulting optimization problems, including compact reformulations that exploit special structures when they appear, a column generation algorithm, and a reformulation of the adversarial problem as a minimum-cost flow problem. Our numerical experiments, based on problems from literature, illustrate (i) that the performance of the smooth uncertainty set model solution is similar to that of the ellipsoidal uncertainty model solution, albeit, it is computed within significantly shorter running times, and (ii) our column-generation algorithm can outperform the classical cutting plane algorithm and dualized reformulation, respectively in terms of solution time and memory consumption.

math.OC↗
arXiv2025

Projecting onto a Capped Rotated Second-Order Cone

We derive a closed-form expression for the projection onto a capped rotated second-order cone -- a convex set that arises in perspective relaxations of nonlinear programs with binary indicator variables. The closed-form solution involves three distinct cases, one of which reduces to the classical projection onto a second-order cone. The remaining two cases yield nontrivial projections, for which we provide necessary and sufficient conditions under which the solution lies on the intersection of the cone and a facet of a box.

math.OC↗
arXiv2024

Robust Radiotherapy Planning with Spatially Based Uncertainty Sets

Radiotherapy treatment planning is a challenging large-scale optimization problem plagued by uncertainty. Following the robust optimization methodology, we propose a novel, spatially based uncertainty set for robust modeling of radiotherapy planning, producing solutions that are immune to unexpected changes in biological conditions. Our proposed uncertainty set realistically captures biological radiosensitivity patterns that are observed using recent advances in imaging, while its parameters can be personalized for individual patients. We exploit the structure of this set to devise a compact reformulation of the robust model. We develop a row-generation scheme to solve real, large-scale instances of the robust model. This method is then extended to a relaxation-based scheme for enforcing challenging, yet clinically important, dose-volume cardinality constraints. The computational performance of our algorithms, as well as the quality and robustness of the computed treatment plans, are demonstrated on simulated and real imaging data. Based on accepted performance measures, such as minimal target dose and homogeneity, these examples demonstrate that the spatially robust model achieves almost the same performance as the nominal model in the nominal scenario, and otherwise, the spatial model outperforms both the nominal and the box-uncertainty models.

math.OC↗
arXiv2020

A chip-scale oscillation-mode optomechanical inertial sensor near the thermodynamical limits

High-precision inertial sensing and gravity sensing are key in navigation, oil exploration, and earthquake prediction. In contrast to prior accelerometers using piezoelectric or electronic capacitance readout techniques, optical readout provides narrow-linewidth high-sensitivity laser detection along with low-noise resonant optomechanical transduction near the thermodynamical limits. Here an optomechanical inertial sensor with 8.2micro-g/Hz^1/2 velocity random walk (VRW) at acquisition rate of 100 Hz and 50.9 micro-g bias instability is demonstrated, suitable for consumer and industrial grade applications, e.g., inertial navigation, inclination sensing, platform stabilization, and/or wearable device motion detection. Driven into optomechanical sustained-oscillation, the slot photonic crystal cavity provides radio-frequency readout of the optically-driven transduction with enhanced 625 microg/Hz sensitivity. Measuring the optomechanically-stiffened oscillation shift, instead of the optical transmission shift, provides a 220x VRW enhancement over pre-oscillation mode detection due to the strong optomechanical transduction. Supported by theory, this inertial sensor operates 2.56x above the thermodynamical limit at small integration times, with 43-dB dynamic range, in a solid-state room-temperature readout architecture.

physics.ins-det↗
arXiv2017

On the Complexity and Approximation of the Maximum Expected Value All-or-Nothing Subset

An unconstrained nonlinear binary optimization problem of selecting a maximum expected value subset of items is considered. Each item is associated with a profit and probability. Each of the items succeeds or fails independently with the given probabilities, and the profit is obtained in the event that all selected items succeed. The objective is to select a subset that maximizes the total value times the product of probabilities of the chosen items. The problem is proven NP-hard by a nontrivial reduction from subset sum. Then we develop a fully polynomial time approximation scheme (FPTAS) for this problem.

cs.CC↗
arXiv2013

Feedback and harmonic locking of slot-type optomechanical oscillators to external low-noise reference clocks

We demonstrate feedback and harmonic locking of chip-scale slot-type optomechanical oscillators to external low-noise reference clocks, with suppressed timing jitter by three orders of magnitude. The feedback and compensation techniques significantly reduce the close-to-carrier phase noise, especially within the locking bandwidth for the integral root-mean-square timing jitter. Harmonic locking via high-order carrier signals is also demonstrated with similar phase noise and integrated root-mean-square timing jitter reduction. The chip-scale optomechanical oscillators are tunable over an 80-kHz range by tracking the reference clock, with potential applications in tunable radio-frequency photonics platforms.

physics.optics↗