Worst-case Performance Guarantees Research Articles

Learning-Augmented Online Algorithm for Two-Level Ski-Rental Problem

In this paper, we study the two-level ski-rental problem, where a user needs to fulfill a sequence of demands for multiple items by choosing one of the three payment options: paying for the on-demand usage (i.e., rent), buying individual items (i.e., single purchase), and buying all the items (i.e., combo purchase). Without knowing future demands, the user aims to minimize the total cost (i.e., the sum of the rental, single purchase, and combo purchase costs) by balancing the trade-off between the expensive upfront costs (for purchase) and the potential future expenses (for rent). We first design a robust online algorithm (RDTSR) that offers a worst-case performance guarantee. While online algorithms are robust against the worst-case scenarios, they are often overly cautious and thus suffer a poor average performance in typical scenarios. On the other hand, Machine Learning (ML) algorithms typically show promising average performance in various applications but lack worst-case performance guarantees. To harness the benefits of both methods, we develop a learning-augmented algorithm (LADTSR) by integrating ML predictions into the robust online algorithm, which outperforms the robust online algorithm under accurate predictions while ensuring worst-case performance guarantees even when predictions are inaccurate. Finally, we conduct numerical experiments on both synthetic and real-world trace data to corroborate the effectiveness of our approach.

Developing New Bounds for the Performance Guarantee of the Jump Neighborhood for Scheduling Jobs on Uniformly Related Machines

This study investigates the worst-case performance guarantee of locally optimal solutions to minimize the total weighted completion time on uniformly related parallel machines. The investigated neighborhood structure is Jump, also called insertion or move. This research focused on establishing the local optimality condition expressed as an inequality and mapping that maps a schedule into an inner product space so that the norm of the mapping is closely related to the total weighted completion time of the schedule. We determine two new upper bounds for the performance guarantee, which take the form of an expression based on parameters that describe the family of instances: the speed of the fastest machine, the speed of the slowest machine, and the number of machines. These new bounds outperform the parametric upper bound previously established in the existing literature and enable a better understanding of the performance of the solutions obtained for the Jump neighborhood in this scheduling problem, according to parameters that describe the family of instances.

Bounds and Heuristics for Multiproduct Pricing

For a large class of demand models that allow for multiple consumer types, we present performance guarantees for simple nonpersonalized pricing heuristics relative to optimal personalized pricing. Our results demonstrate that in a general setting, the effectiveness of pricing along a positive vector depends on how the price vector aligns with optimal personalized price vectors. We propose two positive direction vectors: the “economic” and “robust” directions. The economic direction is a convex combination of the optimal personalized price vectors and aims to do well on average. The robust direction offers the best worst-case performance guarantee. By judiciously selecting pricing directions, our results also provide performance guarantees of simple pricing strategies relative to more sophisticated pricing strategies. In particular, we provide performance guarantees for nonpersonalized, optimal linear pricing relative to optimal nonlinear, personalized pricing. Our research also examines the performance of common heuristics for bundle pricing relative to optimal, personalized, bundle-size pricing. Our experiments show that performance often improves when consumer types are clustered and each cluster is offered a price direction. We compared the performance of the k-means clustering heuristic and the farthest point first clustering heuristic. Our findings indicated that k-means clustering has significantly superior performance on average. This suggests that businesses could potentially benefit from implementing k-means clustering in their pricing strategies. In conclusion, our study offers valuable insights and performance guarantees for various pricing strategies and their relative effectiveness. These findings could inform pricing decisions and potentially lead to improved outcomes for firms. This paper was accepted by Chung-Piaw Teo, optimization. Funding: This work was supported by Collaborative Research Funding (CRF) Hong Kong [Grant (CRF) C6032-21G], RGC Hong Kong [Grant 16502819], and Hong Kong University of Science and Technology. Supplemental Material: The data files are available at https://doi.org/10.1287/mnsc.2021.04173 .

Energy-Aware, Collision-Free Information Gathering for Heterogeneous Robot Teams

This article considers the problem of safely coordinating a team of sensor-equipped robots to reduce uncertainty about a dynamical process, where the objective tradeoffs information gain and energy cost. Optimizing this tradeoff is desirable, but leads to a nonmonotone objective function in the set of robot trajectories. Therefore, common multirobot planners based on coordinate descent lose their performance guarantees. Furthermore, methods that handle nonmonotonicity lose their performance guarantees when subject to interrobot collision avoidance constraints. As it is desirable to retain both the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">performance guarantee</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">safety guarantee</i> , this work proposes a hierarchical approach with a distributed planner that uses local search with a worst-case performance guarantees and a decentralized controller based on control barrier functions that ensures safety and encourages timely arrival at sensing locations. Via extensive simulations, hardware-in-the-loop tests, and hardware experiments, we demonstrate that the proposed approach achieves a better tradeoff between sensing and energy cost than coordinate-descent-based algorithms.

Inventory replenishment decisions with uncertain price and demand

This paper studies the multiple period inventory replenishment problem for a capacitated warehouse when procurement prices and market demands are both uncertain, with the goal to minimise the procuring and holding costs. We analyse this problem under the framework of competitive analysis, where neither probabilistic distributions nor sets are available to characterise the unknown price and demand parameters. An efficient online real-time replenishment algorithm is developed, which is free of any distribution assumption, and the decisions are made based entirely on past and present information. We derive an instance-independent competitive ratio of the algorithm, which provides a worst-case theoretical performance guarantee, and shows that the proposed algorithm performs well for situations with high volatility uncertainty over time and is naturally risk-averse. Finally, a set of numerical experiments further verifies the effectiveness of the algorithm.

Mechanism Design with Predictions: An Annotated Reading List

A surge of recent work has focused on analyzing the performance of algorithms guided by predictions, aiming to enhance their worst-case performance guarantees with improved guarantees when the predictions are accurate. This "learning-augmented" framework was recently also extended to mechanism design settings involving strategic agents and we provide an overview of these results.

Secretary and online matching problems with machine learned advice

The classic analysis of online algorithms, due to its worst-case nature, can be quite pessimistic when the input instance at hand is far from worst-case. In contrast, machine learning approaches shine in exploiting patterns in past inputs in order to predict the future. However, such predictions, although usually accurate, can be arbitrarily poor. Inspired by a recent line of work, we augment three well-known online settings with machine learned predictions about the future, and develop algorithms that take these predictions into account. In particular, we study the following online selection problems: (i) the classic secretary problem, (ii) online bipartite matching and (iii) the graphic matroid secretary problem. Our algorithms still come with a worst-case performance guarantee in the case that predictions are subpar while obtaining an improved competitive ratio (over the best-known classic online algorithm for each problem) when the predictions are sufficiently accurate. For each algorithm, we establish a trade-off between the competitive ratios obtained in the two respective cases.

Autonomous Recharging and Flight Mission Planning for Battery-Operated Autonomous Drones

Unmanned aerial vehicles (UAVs), commonly known as drones, are being increasingly deployed throughout the globe as a means to streamline monitoring, inspection, mapping, and logistic routines. When dispatched on autonomous missions, drones require an intelligent decision-making system for trajectory planning and tour optimization. Given the limited capacity of their onboard batteries, a key design challenge is to ensure the underlying algorithms can efficiently optimize the mission objectives along with recharging operations during long-haul flights. With this in view, the present work undertakes a comprehensive study on automated tour management systems for an energy-constrained drone: (1) We construct a machine learning model that estimates the energy expenditure of typical multi-rotor drones while accounting for real-world aspects and extrinsic meteorological factors. (2) Leveraging this model, the joint program of flight mission planning and recharging optimization is formulated as a multi-criteria Asymmetric Traveling Salesman Problem (ATSP), wherein a drone seeks for the time-optimal energy-feasible tour that visits all the target sites and refuels whenever necessary. (3) We devise an efficient approximation algorithm with provable worst-case performance guarantees and implement it in a drone management system, which supports real-time flight path tracking and re- computation in dynamic environments. (4) The effectiveness and practicality of the proposed approach are validated through extensive numerical simulations as well as real-world experiments. Note to Practitioners—This study is stimulated by the need for developing pragmatic and provably efficient automated tour management systems for UAVs deployed on energy-constrained, long-distance flight missions. As such, UAVs provide a nifty platform for facilitating environmental monitoring, disaster management, transport of medical supplies, as well as expediting last-mile deliveries. However, existing path planners generally fall short of capturing several crucial aspects, such as detailed power consumption model (e.g., factoring in payload, wind speed and direction) or performance guarantees, potentially leading to underutilized or infeasible routing decisions. To address these issues, the present work proposes a theoretically-backed routing approach with a certifiable degree of optimality and develops an effective, practical power consumption evaluation model for multi-rotor UAVs, verified on multiple drone models.

QoE-Driven Adaptive Streaming for Point Clouds

With increasing popularity of virtual reality and augmented reality, application of point clouds is in critical demand as it enables users to freely navigate in an immersive scene with six degrees of freedom. However, point clouds usually comprise large amounts of data, and are thus difficult to stream in bandwidth-constrained networks. It is therefore important, yet challenging, to efficiently stream the resource-intensive point clouds, such that the user's quality of experience (QoE) is guaranteed on a high-level but with a low bandwidth consumption. To this end, we propose a QoE-driven adaptive streaming approach for the tile-based point cloud transmission, to maximize the user's QoE while reducing the transmission redundancy. By exploiting the perspective projection, we specifically model the QoE of a 3D tile as a function of the bitrate of its representation, user's view frustum and spatial position, occlusion between tiles, and the resolution of rendering device. Based on this QoE model, we then formulate the QoE-optimized rate adaptation problem as a multiple-choice knapsack problem, which allocates bitrates for different tiles under a given transmission capacity. It is equivalently converted to a submodular function maximization problem subject to knapsack constraints, and solved by a practical greedy-based algorithm with a theoretical worst-case performance guarantee. The proposed algorithm is able to achieve a near-optimal performance, but with a very low computational complexity. Experimental results further demonstrate superiority of the proposed rate adaptation algorithm over existing schemes, in terms of both user's visual quality and transmission efficiency.

The Online Knapsack Problem with Departures

The online knapsack problem is a classic online resource allocation problem in networking and operations research. Its basic version studies how to pack online arriving items of different sizes and values into a capacity-limited knapsack. In this paper, we study a general version that includes item departures, while also considering multiple knapsacks and multi-dimensional item sizes. We design a threshold-based online algorithm and prove that the algorithm can achieve order-optimal competitive ratios. Beyond worst-case performance guarantees, we also aim to achieve near-optimal average performance under typical instances. Towards this goal, we propose a data-driven online algorithm that learns within a policy-class that guarantees a worst-case performance bound. In trace-driven experiments, we show that our data-driven algorithm outperforms other benchmark algorithms in an application of online knapsack to job scheduling for cloud computing.

Simultaneously Transmitting and Reflecting Reconfigurable Intelligent Surface (STAR-RIS) Assisted UAV Communications

A novel air-to-ground communication paradigm is conceived, where an unmanned aerial vehicle (UAV)-mounted base station (BS) equipped with multiple antennas sends information to multiple ground users (GUs) with the aid of a simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS). In contrast to the conventional RIS whose main function is to reflect incident signals, the STAR-RIS is capable of both transmitting and reflecting the impinging signals from either side of the surface, thereby leading to full-space 360 degree coverage. However, the transmissive and reflective capabilities of the STAR-RIS require more complex transmission/reflection coefficient design. Therefore, in this work, a sum-rate maximization problem is formulated for the joint optimization of the UAV’s trajectory, the active beamforming at the UAV, and the passive transmission/reflection beamforming at the STAR-RIS. This cutting-edge optimization problem is also subject to the UAV’s flight safety, to the maximum flight duration constraint, as well as to the GUs’ minimum data rate requirements. Given the unknown locations of obstacles prior to the UAV’s flight, we provide an online decision making framework employing reinforcement learning (RL) to simultaneously adjust both the UAV’s trajectory as well as the active and passive beamformer. To enhance the system’s robustness against the associated uncertainties caused by limited sampling of the environment, a novel “distributionally-robust” RL (DRRL) algorithm is proposed for offering an adequate worst-case performance guarantee. Our numerical results unveil that: 1) the STAR-RIS assisted UAV communications benefit from significant sum-rate gain over the conventional reflecting-only RIS; and 2) the proposed DRRL algorithm achieves both more stable and more robust performance than the state-of-the-art RL algorithms.

Online Multipath Routing via Multiplicative Weight Update

The multiplicative weight update (MWU) schema is a powerful tool to design online routing algorithms. However, even the latest research still employs or incorporates the inefficient online single-path routing (OSR) algorithm. In this article, we design an online multipath routing (OMR) algorithm via the MWU schema in order to exploit the network path diversity. Compared to OSR and comparative multipath solutions, OMR has the same worst-case performance guarantee in theory and also has significant performance advantage in practice.

Expert-Calibrated Learning for Online Optimization with Switching Costs

We study online convex optimization with switching costs, a practically important but also extremely challenging problem due to the lack of complete offline information. By tapping into the power of machine learning (ML) based optimizers, ML-augmented online algorithms (also referred to as expert calibration in this paper) have been emerging as state of the art, with provable worst-case performance guarantees. Nonetheless, by using the standard practice of training an ML model as a standalone optimizer and plugging it into an ML-augmented algorithm, the average cost performance can be highly unsatisfactory. In order to address the "how to learn" challenge, we propose EC-L2O (expert-calibrated learning to optimize), which trains an ML-based optimizer by explicitly taking into account the downstream expert calibrator. To accomplish this, we propose a new differentiable expert calibrator that generalizes regularized online balanced descent and offers a provably better competitive ratio than pure ML predictions when the prediction error is large. For training, our loss function is a weighted sum of two different losses --- one minimizing the average ML prediction error for better robustness, and the other one minimizing the post-calibration average cost. We also provide theoretical analysis for EC-L2O, highlighting that expert calibration can be even beneficial for the average cost performance and that the high-percentile tail ratio of the cost achieved by EC-L2O to that of the offline optimal oracle (i.e., tail cost ratio) can be bounded. Finally, we test EC-L2O by running simulations for sustainable datacenter demand response. Our results demonstrate that EC-L2O can empirically achieve a lower average cost as well as a lower competitive ratio than the existing baseline algorithms.

Surface Code Compilation via Edge-Disjoint Paths

We provide an efficient algorithm to compile quantum circuits for fault-tolerant execution. We target surface codes, which form a 2D grid of logical qubits with nearest-neighbor logical operations. Embedding an input circuit's qubits in surface codes can result in long-range two-qubit operations across the grid. We show how to prepare many long-range Bell pairs on qubits connected by edge-disjoint paths of ancillas in constant depth that can be used to perform these long-range operations. This forms one core part of our Edge-Disjoint Paths Compilation (EDPC) algorithm, by easily performing many parallel long-range Clifford operations in constant depth. It also allows us to establish a connection between surface code compilation and several well-studied edge-disjoint paths problems. Similar techniques allow us to perform non-Clifford single-qubit rotations far from magic state distillation factories. In this case, we can easily find the maximum set of paths by a max-flow reduction, which forms the other major part of EDPC. EDPC has the best asymptotic worst-case performance guarantees on the circuit depth for compiling parallel operations when compared to related compilation methods based on swaps and network coding. EDPC also shows a quadratic depth improvement over sequential Pauli-based compilation for parallel rotations requiring magic resources. We implement EDPC and find significantly improved performance for circuits built from parallel cnots, and for circuits which implement the multi-controlled $X$ gate.

RIS-Aided Ground-Aerial NOMA Communications: A Distributionally Robust DRL Approach

A reconfigurable intelligent surface (RIS) aided air-to-ground uplink non-orthogonal transmission framework is investigated for next generation multiple access. Occupying the same spectrum resource, unmanned aerial vehicle (UAV) users and ground users (GUs) are connected to terrestrial cellular networks via the uplink non-orthogonal multiple access (NOMA) protocol. As the flight safety is important for employing UAVs in civil airspace, the collision avoidance mechanism has to be considered during the flight. Therefore, a joint optimization problem of the UAV trajectory design, RIS configuration, and uploading power control is formulated for maximizing the network sum rate, while ensuring the UAV’s fight safety and satisfying the minimum data rate requirements of both the UAV and GU. The resultant problem is a sequential decision making one across multiple coherent time slots. Besides, the unknown locations of obstacles bring uncertainties into the decision making process. To tackle this challenging problem, a sample-efficient deep reinforcement learning (DRL) algorithm is proposed to optimize the UAV trajectory, RIS configuration, and power control simultaneously. Moreover, considering the ambiguous uncertainties in the environment, a distributionally robust DRL algorithm is further proposed to provide the worst-case performance guarantee. Numerical results demonstrate that the two proposed DRL algorithms outperform the conventional ones in terms of learning efficiency and robustness. It is also shown that the network sum rate is significantly improved by the proposed RIS-NOMA scheme compared to the conventional RIS-orthogonal multiple access (OMA) scheme and the case where no RIS is deployed.

Performance guarantee of the jump neighborhood for scheduling jobs on uniformly related machines

We study the worst case performance guarantee of locally optimal solutions for the problem of scheduling jobs on uniformly related parallel machines with the objective of minimizing the total weighted completion time. The quality of locally optimal solutions under the jump neighborhood is analyzed, which consists of iteratively moving a single job from one machine to another, improving the total weighted completion time in each iteration and stopping once improvement is no longer possible. We propose an upper bound for the total weighted completion time for the solutions obtained by this local search, and upper and lower bounds for the performance guarantee of the obtained locally optimal solutions. Additionally, the case of identical parallel machines is analyzed.

Towards accelerated greedy sampling and reconstruction of bandlimited graph signals

We study the problem of sampling and reconstructing spectrally sparse graph signals where the objective is to select a subset of nodes of prespecified cardinality that ensures interpolation of the original signal with the lowest possible reconstruction error. This task is of critical importance in Graph signal processing (GSP) and while existing methods generally provide satisfactory performance, they typically entail a prohibitive computational cost when it comes to the study of large-scale problems. Thus, there is a need for accelerated and efficient methods tailored for high-dimensional and large-scale sampling and reconstruction tasks. To this end, we first consider a non-Bayesian scenario and propose an efficient iterative node sampling procedure that in the noiseless case enables exact recovery of the original signal from the set of selected nodes. In the case of noisy measurements, a bound on the reconstruction error of the proposed algorithm is established. Then, we consider the Bayesian scenario where we formulate the sampling task as the problem of maximizing a monotone weak submodular function, and propose a randomized-greedy algorithm to find a sub-optimal subset of informative nodes. We derive worst-case performance guarantees on the mean-square error achieved by the randomized-greedy algorithm for general non-stationary graph signals.

SR-WMN: Online Network Throughput Optimization in Wireless Mesh Networks With Segment Routing

In this letter, we for the first time introduce segment routing (SR) to wireless mesh networks (WMNs) under the primary interference model, and try to obtain the upper bound of maximum network throughput from an online routing perspective. We first propose an SR-enabled WMN (SR-WMN) architecture. We then develop an online primal-dual algorithm with guaranteed worst case performance bound. We finally show via simulation that the proposed solution tends to be close to the optimum by configuring just over half of the wireless mesh routers (WMRs) to support SR.

Distribution-Free Pricing

Problem definition: We study a monopolistic robust pricing problem in which the seller does not know the customers’ valuation distribution for a product but knows its mean and variance. Academic/practical relevance: This minimal requirement for information means that the pricing managers only need to be able to answer two questions: How much will your targeted customers pay on average? To measure your confidence in the previous answer, what is the standard deviation of customer valuations? Methodology: We focus on the maximin profit criterion and derive distribution-free upper and lower bounds on the profit function. Results: By maximizing the tight profit lower bound, we obtain the optimal robust price in closed form as well as its distribution-free, worst-case performance bound. We then extend the single-product result to study the robust pure bundle pricing problem where the seller only knows the mean and variance of each product, and we provide easily verifiable, distribution-free, sufficient conditions that guarantee the pure bundle to be more robustly profitable than à la carte (i.e., separate) sales. We further derive a distribution-free, worst-case performance guarantee for a heuristic scheme in which customers choose between buying either a single product or a pure bundle. Moreover, we generalize separate sales and pure bundling to a scheme called clustered bundling that imposes a price for each part (i.e., cluster) of a partition of all products and allows customers to choose one or multiple parts (i.e., clusters), and we provide various algorithms to compute clustered bundling heuristics. In parallel, most of our results hold for the minimax relative regret criterion as well. Managerial implications: The robust price for a single product is in closed form under the maximin profit or minimax relative regret criterion and hence, is easily computable. Its interpretation can be easily explained to pricing managers. We also provide efficient algorithms to compute various mixed bundling heuristics for the multiproduct problem.

Probability-Based Online Algorithm for Switch Operation of Energy Efficient Data Center

The huge amount of energy consumed by the data centers around the world every year motivates the cloud service providers to operate data centers in a more energy efficient way. A promising solution is to turn off the idle servers, which, however, may be turned on later, incurring a significant startup cost. The problem turns to dynamically provisioning the workload, and cutting down the energy cost which includes the power to support the running of data center and the startup cost. Different from previous studies that usually consider the worst case performance guarantee when designing online algorithms, this paper considers the average case which is more practical. We propose a simple online algorithm based on the expectation of job interval of workload, which is proven to be optimal for exponential and uniform distributions and achieves tight competitive ratio $\frac{e}{e-1}$ and $\frac{4}{3}$ respectively for them. Simulations using the synthetic data verify our theoretical analysis. Numerical results employing Google's data center workload trace demonstrate that the proposed algorithm outperforms the worst case based algorithm in terms of operation cost reduction.