Worst-case Performance Guarantees Research Articles

1.79-Approximation Algorithms for Continuous Review Single-Sourcing Lost-Sales and Dual-Sourcing Inventory Models

Stochastic inventory systems with lead times are often challenging to optimize, including single-sourcing lost-sales and dual-sourcing systems. Recent numerical results suggest that capped policies demonstrate superior performance over existing heuristics. However, the superior performance lacks a theoretical foundation. In “1.79-Approximation Algorithms for Continuous Review Single-Sourcing Lost-Sales and Dual-Sourcing Inventory Models,” the author provides a theoretical foundation for this phenomenon in two classical inventory models. First, in a continuous review lost-sales model with lead times and Poisson demand, he proves that a capped base-stock policy has a worst-case performance guarantee of 1.79 by conducting an asymptotic analysis under a large penalty cost and lead time. Second, in a more complex continuous review dual-sourcing model with general lead times and Poisson demand, he proves that a similar capped dual-index policy has a worst-case performance guarantee of 1.79 under large lead time and ordering cost differences. The results provide a deeper understanding of the superior numerical performance of capped policies and present a new approach to proving worst-case performance guarantees of simple policies in hard inventory problems.

MaxCut quantum approximate optimization algorithm performance guarantees for p>1

We obtain worst case performance guarantees for $p=2$ and $3$ QAOA for MAXCUT on uniform 3-regular graphs. Previous work by Farhi et al obtained a lower bound on the approximation ratio of $0.692$ for $p=1$. We find a lower bound of $0.7559$ for $p=2$, where worst case graphs are those with no cycles $\leq 5$. This bound holds for any 3 regular graph evaluated at particular fixed parameters. We conjecture a hierarchy for all $p$, where worst case graphs have with no cycles $\leq 2p+1$. Under this conjecture, the approximation ratio is at least $0.7924$ for all 3 regular graphs and $p=3$. In addition, using a simple indistinguishability argument we find an upper bound on the worst case approximation ratio for all $p$, which indicates classes of graphs for which there can be no quantum advantage for at least $p<6$.

Worst-case traversal time analysis of TSN with multi-level preemption

Abstract To achieve low latency transmission of time-sensitive flows in Ethernet networks, the IEEE introduced the IEEE 802.1Qbu Standard, which specifies a 1-level preemption scheme for IEEE 802.1 networks. The specification allows the suspension of a preemptable frame prior to its completion for the speedy transmission of an express frame; but any other preemptable frame cannot be transmitted before the completion of the already preempted frame. While this approach improves the performance of express frames, the performance is negatively impacted in scenarios where the number of express frames is high. Another limitation is the fact that preemptable frames with timing requirements can suffer long blocking periods due to the non-preemptive service of frames in the same category. This is irrespective of the individual priority level of each frame. Recently, a multi-level preemption scheme has been proposed to circumvent these limitations. The work focused on the feasibility and implementation requirement of such an approach, but a formal analysis of the worst-case performance guarantees under the proposed scheme was not provided. In this paper, we fill this gap by presenting the aforementioned analysis of the TSN IEEE 802.1Qbu networks under the multi-level preemption assumption. Evaluation is performed with a realistic automotive use-case and the results showcase an improvement up to 53.07% for preemptable frames with firm timing requirements.

An Instance Reservation Framework for Cost Effective Services in Geo-Distributed Data Centers

Infrastructure-as-a-Service clouds in geo-distributed data centers offer various pricing options, including on-demand and reserved instances, which provide an elastic and cost-effective infrastructure to support High Performance Computing (HPC) applications. In this paper, we propose an instance reservation based cloud service framework, modeling the cost-minimizing reservation decision issue as an NP-hard integer programming problem for distributed data centers. To ease its computation complexity, two algorithms are proposed to minimize the HPC service cost with the worst-case performance guarantees: an offline heuristic-greedy algorithm, and a rolling-horizon based online algorithm when only short-term demand prediction is available. Facing fluctuating demands, instance reservation in a single data center may incur the highly underutilized capacity. To address this issue for further cost reduction, we extend the scheme with a novel cloud broker federation based resource sharing mechanism, reallocating already reserved but unused instances to computation-intensive and short-lived tasks for continuous execution without interruption. Extensive evaluations driven by large-scale trace-based datasets demonstrate that the proposed mechanism can effectively handle large volumes of service requests, saving considerable service costs with higher reservation resource utilization.

Competitive buffer management for packets with latency constraints

Modern datacenters are increasingly required to deal with latency-sensitive applications. Incorporation of multiple traffic characteristics (e.g., packet values and required processing requirements) significantly increases the complexity of buffer management policies. In this context two major questions arise: how to represent the latency in desired objectives and how to provide guarantees for buffer management policies that would hold across a wide variety of traffic patterns. In this work, we consider a single queue buffering architecture, where every incoming packet is prepended with intrinsic value, required processing, and slack (offset from the arrival time during which this packet should be transmitted); the buffer size is implicitly bounded by slack values. Our goal is to maximize a total transmitted value (weighted throughput). In these settings, we study worst-case performance guarantees of the proposed online algorithms by means of competitive analysis whose effectiveness is compared versus an optimal clairvoyant offline algorithm. We show non-constant general lower bounds that hold for arbitrary slack values and for slacks that are additively separated from processing requirements; for the case of a multiplicative separation, we present a novel buffer management policy SPQ (stack with priority queue) and show that it is at most 3-competitive. Our theoretical results are supported by a comprehensive evaluation study on CAIDA traces.

Approximation Schemes for the Joint Inventory Selection and Online Resource Allocation Problem

In this paper, we introduce and study the joint inventory and online resource allocation problem, which is characterized by two sequential sets of decisions that are irrevocably linked. First, a decision maker must select starting inventory levels for a set of available resources. Subsequently, the decision maker must match arriving customers to available resources in an online fashion so as to maximize expected reward. We fi rst study the problem in its most general form before focusing on a specific version that arises at Anheuser Busch InBev (ABI). This particular application of our general setting is referred to as the ABI Trailer Problem, and it considers how ABI ships its beer to vendors via third party delivery trucks. Specifically, in this problem, ABI must select the weights and inventory levels of preloaded trailers of beer, which are then matched in an online fashion to arriving third party delivery trucks. For the general setting, we develop simple and easy-to-implement approaches that come with robust worst-case performance guarantees. For the ABI setting, we reveal a simplifying structural property related to the optimal matching policy, which gives rise to a natural adaptation of our original approach. We test the efficacy of these policies through extensive numerical experiments, where we find that our approaches are either near-optimal or improve upon state-of-the-art benchmarks. In particular, using a data set from ABI, we are able to generate instances of the ABI Trailer Problem, on which our algorithm has the potential to yield revenue improvements in the range of millions of dollars per year.

Performance guarantees of local search for minsum scheduling problems

We study the worst-case performance guarantee of locally optimal solutions for the problem of minimizing the total weighted and unweighted completion time on parallel machine environments. Our method makes use of a mapping that maps a schedule into an inner product space so that the norm of the mapping is closely related to the cost of the schedule. We apply the method to study the most basic local search heuristics for scheduling, namely jump and swap, and establish their worst-case performance in the case of unrelated, restricted related and restricted identical machines.

Submodularity in Action: From Machine Learning to Signal Processing Applications

Submodularity is a discrete domain functional property that can be interpreted as mimicking the role of well-known convexity/concavity properties in the continuous domain. Submodular functions exhibit strong structure that lead to efficient optimization algorithms with provable near-optimality guarantees. These characteristics, namely, efficiency and provable performance bounds, are of particular interest for signal processing (SP) and machine learning (ML) practitioners, as a variety of discrete optimization problems are encountered in a wide range of applications. Conventionally, two general approaches exist to solve discrete problems: 1) relaxation into the continuous domain to obtain an approximate solution or 2) the development of a tailored algorithm that applies directly in the discrete domain. In both approaches, worst-case performance guarantees are often hard to establish. Furthermore, they are often complex and thus not practical for large-scale problems. In this article, we show how certain scenarios lend themselves to exploiting submodularity for constructing scalable solutions with provable worst-case performance guarantees. We introduce a variety of submodular-friendly applications and elucidate the relation of submodularity to convexity and concavity, which enables efficient optimization. With a mixture of theory and practice, we present different flavors of submodularity accompanying illustrative real-world case studies from modern SP and ML. In all of the cases, optimization algorithms are presented along with hints on how optimality guarantees can be established.

Robust Active Mirror Control Based on Hybrid Sensing for Spacecraft Line-of-Sight Stabilization

Modern space-observation missions demand stringent pointing requirements that motivated a significant amount of research on the topic of microvibration isolation and line-of-sight stabilization systems. While disturbances can be reduced by mounting some of the noisy equipment on various isolation platforms, residual vibrations can still propagate through and be amplified by the flexible structure of the spacecraft. In order to alleviate these issues, the line of sight must also be actively controlled at the payload level. However, such systems typically have to rely solely on low-frequency sensors based on image processing algorithms. The goal of this article is to present a model-based control methodology that can increase the bandwidth of such systems by making use of additional rate sensors mounted on the main disturbance elements impacting the optical path. Following a comprehensive model identification and uncertainty quantification part, the robust control strategy is designed to account for plant uncertainty and provide formal worst case performance guarantees. Excellent agreement between theoretical prediction and experimental results are obtained on a test bench developed at the European Space Agency.

A Distributed Framework for Task Offloading in Edge Computing Networks of Arbitrary Topology

An important issue in an edge computing (EC) network is to increase the utilities of the end users concurrently accessing the computation resources. In this paper, we consider the task offloading in EC-enabled networks where the end users efficiently utilize the dispersed computation and communication resources in a multi-path multi-hop manner. We propose a binary optimization framework that generalizes multi-hop wireless EC task offloading as jointly making decisions of server selecting and traffic routing in networks of arbitrary topology ( JoSRAT ). We further develop an approximation algorithm JoSRAT that enables for a fully distributed implementation together with the worst-case performance guarantees. Interestingly, our proposed distributed algorithm achieves nearly optimal in the numerical evaluations, significantly outperforming the worst-case guarantees. The proposed algorithm also outperforms a widely-used heuristic, i.e., First Fit, in terms of computational time complexity, indicating the superior capability of the proposed framework.

Learning Product Rankings Robust to Fake Users

In many online platforms, customers' decisions are substantially influenced by product rankings as most customers only examine a few top-ranked products. Concurrently, such platforms also use the same data corresponding to customers' actions to learn how these products must be ranked or ordered. These interactions in the underlying learning process, however, may incentivize sellers to artificially inflate their position by employing fake users, as exemplified by the emergence of click farms. Motivated by such fraudulent behavior, we study the ranking problem of a platform that faces a mixture of real and fake users who are indistinguishable from one another. We first show that existing learning algorithms---that are optimal in the absence of fake users---may converge to highly sub-optimal rankings under manipulation by fake users. To overcome this deficiency, we develop efficient learning algorithms under two informational environments: in the first setting, the platform is aware of the number of fake users, and in the second setting, it is agnostic to the number of fake users. For both these environments, we prove that our algorithms converge to the optimal ranking, while being robust to the aforementioned fraudulent behavior; we also present worst-case performance guarantees for our methods, and show that they significantly outperform existing algorithms. At a high level, our work employs several novel approaches to guarantee robustness such as: (i) constructing product-ordering graphs that encode the pairwise relationships between products inferred from the customers' actions; and (ii) implementing multiple levels of learning with a judicious amount of bi-directional cross-learning between levels. Overall, our results indicate that online platforms can effectively combat fraudulent users without incurring large costs by designing new learning algorithms that guarantee efficient convergence even when the platform is completely oblivious to the number and identity of the fake users.

Dynamic Stochastic Matching Under Limited Time

In centralized matching markets, such as kidney exchange schemes and car-pooling platforms, new participants constantly enter the market and remain available for potential matches during a limited period of time. To reach an efficient allocation, the timing'' of the matching decisions is a critical aspect of the platform's operations. There is a fundamental tradeoff between increasing market thickness and mitigating the risk that participants abandon the market. Nonetheless, the dynamic properties of matching markets have been mostly overlooked in the classic algorithmic literature. In this paper, we introduce a general dynamic matching model over edge-weighted graphs, where the agents' arrivals and abandonments are stochastic and heterogeneous. While the resulting Markov decision process is computationally intractable, we design new simple matching algorithms that admit strong worst-case performance guarantees. We devise a 3-approximation algorithm for the cost-minimization problem on graphs satisfying a metric-like property. In contrast, we show that the performance of widely used batching algorithms can be arbitrarily bad. We further devise a (e-1)/(2e)-approximation algorithm for the reward-maximization problem on arbitrary bipartite graphs. Our approach involves the development of a host of techniques, including linear programming benchmarks, value function approximations, and proxies for continuous-time Markov chains, that could be of broader interest. In extensive experiments, we simulate the matching operations of a car-pooling platform using real-world taxi demand data. The newly developed algorithms have the potential to significantly improve cost efficiency against widely used batching algorithms.

A 1.79-approximation Algorithm for a Continuous Review Lost-Sales Inventory Model

Stochastic inventory systems with lead times are often challenging to optimize, including single-sourcing lost-sales and dual-sourcing inventory systems. Recent numerical results have suggested that capped policies demonstrate superior performance over existing heuristics. However, the superior performance lacks a theoretical foundation, and why such policies generally perform so well remains a major open question. In this paper, we provide a theoretical foundation for this phenomenon in two classical inventory models with lead times. First, in a continuous review lost-sales inventory model with lead times and Poisson demand, we prove that a so-called capped base-stock policy has a worst-case performance guarantee of 1.79, by conducting an asymptotic analysis under a large penalty cost and lead time. Second, we extend the analysis to a more complex continuous review dual-sourcing inventory model with general lead times and Poisson demand, and also prove that a so-called capped dual-index policy has a worst-case performance guarantee of 1.79 under large lead time and ordering cost differences. Our results provide a deeper understanding of the superior numerical performance of capped policies, and present a new approach to proving worst-case performance guarantees of simple policies in hard inventory problems.

On fractional cut covers

Given an undirected graph, a minimum cut cover is a collection of cuts covering the whole set of edges and having minimum cardinality. This paper is dedicated to the fractional version of this problem where a fractional weight is computed for each cut such that, for each edge, the sum of the weights of all cuts containing it is no less than 1, while the sum of all weights is minimized. The fractional cover is computed for different graph classes among which are the weakly bipartite graphs. Efficient algorithms are described to compute lower and upper bounds with worst-case performance guarantees. A general randomized approach is also presented giving new insights into Goemans and Williamson’s algorithm for the maximum cut problem. Some numerical experiments are included to assess the quality of bounds.

Approaches for Assigning Offsets to Signals for Improving Frame Packing in CAN-FD

Controller area network (CAN) is a widely used protocol that allows communication among electronic control units (ECUs) in automotive electronics. It was extended to CAN with flexible data-rate (CAN-FD) to meet the increasing demand for bandwidth generated by the growing number of features in modern automobiles. The signal-to-frame packing problem has been studied in the literature for both CAN and CAN-FD. In this paper, we propose and formulate the signal offset assignment problem (SOAP) in CAN-FD to improve the bus utilization during frame packing. We propose two algorithmic themes to solve SOAP and establish their worst case performance guarantees. The first is a general approximation framework (GAF) which can use any approximation algorithm for the makespan minimization problem (MMP) in multiprocessor systems. Its performance guarantee is the product of the performance guarantee of the MMP algorithm and the number of distinct periods in the frame. The second is a 2-D strip packing-based framework (2DSPF) which uses the bottom left fill algorithm for 2-D strip packing. The performance guarantee is $2{G}$ , where ${G}$ is the minimum number of groups into which the set of signals can be partitioned so that the periods of the signals in the same group form a geometric series. The experimental results for GAF and 2DSPF indicate that by carefully assigning offsets for signals in frame packing schemes, one can achieve about 10.83% improvement in bus utilization in CAN-FD systems.

Data-Driven Decisions for Problems with an Unspecified Objective Function

This study develops a data-driven approach to solve constrained optimization problems in which the decision maker does not have an analytic form for the objective function but knows what decision variables affect the function. The approach makes direct use of the available data, rather than first using the data to estimate the objective function and then solving the problem as a traditional optimization problem. The difficulty in first estimating the unknown objective function is that the decision maker needs to have sufficient knowledge of its properties that are necessary to guide the estimation process. Thus, our approach is appropriate for situations where such structural knowledge is absent, either because the domain is very complex or because the knowledge is deliberately hidden by a partner firm that has a vested interest in the outcome of the decision. Our approach comes with a worst-case performance guarantee that improves with the characteristics (size, pervasiveness) of the available data. We illustrate our technique on a traffic-stream mixing problem encountered by a supply side Internet advertising network that wishes to optimize the click revenue earned from ads. A head-to-head comparison (with the existing method used) on real data shows a significant increase (≥10%, on average) in the revenue. We also demonstrate the value of our approach under more general conditions. The online supplement is available at https://doi.org/10.1287/ijoc.2018.0818 .

A Distributed Computation Offloading Strategy in Small-Cell Networks Integrated With Mobile Edge Computing

Mobile edge computing is conceived as an appealing technology to enhance cloud computing capability of mobile devices (MDs) at the edge of the networks. Although some researchers use the technology to address the intensive tasks’ high computation needs of MDs in small-cell networks (SCNs), most of them ignore considering the interests interaction between small cells and MDs. In this paper, we study a distributed computation offloading strategy for a multi-device and multi-server system based on orthogonal frequency-division multiple access in SCNs. First, to satisfy the interest requirements of different MDs and analyze the interactions among multiple small cells, we formulate a distributed overhead minimization problem, aiming at jointly optimizing energy consumption and latency of each MD. Second, to ensure the individuals of different MDs, we formulate the proposed overhead minimization problem as a strategy game. Then, we prove the strategy game is a potential game by the feat of potential game theory. Moreover, the potential game-based offloading algorithm is proposed to reach a Nash equilibrium. In addition, to guarantee the performance of the designed algorithm, we consider the lower bound of iteration times to derive the worst case performance guarantee. Finally, the simulation results corroborate that the proposed algorithm can effectively minimize the overhead of each MD compared with different other existing algorithms.

EGRank: An exponentiated gradient algorithm for sparse learning-to-rank

This paper focuses on the problem of sparse learning-to-rank, where the learned ranking models usually have very few non-zero coefficients. An exponential gradient algorithm is proposed to learn sparse models for learning-to-rank, which can be formulated as a convex optimization problem with the ℓ1 constraint. Our proposed algorithm has a competitive theoretical worst-case performance guarantee, that is, we can obtain an ϵ-accurate solution after O(1ϵ) iterations. An early stopping criterion based on Fenchel duality is proposed to make the algorithm be more efficient in practice. Extensive experiments are conducted on some benchmark datasets. The results demonstrate that a sparse ranking model can significantly improve the accuracy of ranking prediction compared to dense models, and the proposed algorithm shows stable and competitive performance compared to several state-of-the-art baseline algorithms.

Fast algorithms for fragmentable items bin packing

Bin packing with fragmentable items is a variant of the classic bin packing problem where items may be cut into smaller fragments. The objective is to minimize the number of item fragments, or equivalently, to minimize the number of cuts, for a given number of bins. Models based on packing fragmentable items are useful for representing finite shared resources. In this article, we present improvements to approximation and metaheuristic algorithms to obtain an optimality-preserving optimization algorithm with polynomial complexity, worst-case performance guarantees and parametrizable running time. We also present a new family of fast lower bounds and prove their worst-case performance ratios. We evaluate the performance and quality of the algorithm and the best lower bound through a series of computational experiments on representative problem instances. For the studied problem sets, one consisting of 180 problems with up to 20 items and another consisting of 450 problems with up to 1024 items, the lower bound performs no worse than 5 / 6. For the first problem set, the algorithm found an optimal solution in 92 % of all 1800 runs. For the second problem set, the algorithm found an optimal solution in 99 % of all 4500 runs. No run lasted longer than 220 ms.

A Switch in Time Saves the Dime: A Model to Reduce Rental Cost in Cloud Computing

The need to continually reduce operating costs while meeting computational needs, are goals common to all modern organizations that use cloud computing. We study the problem of selecting computing resources with the goal to minimize the total rental cost of completing a computing task in the presence of a time constraint. The problem is formulated as a scheduling problem that assigns computing resources to time periods of the planning horizon (time available to complete a single computing task). This (NP-hard) preemptive-resume type scheduling problem – new to the scheduling literature – has not been carefully addressed in practice to provide an implementable solution. Typically, the approach taken in practice is to use a single resource (a single virtual machine instance, or a cluster of identical virtual machine instances) to complete a computing task. The main insight of this study is that rather than completing a computing task using a single computing resource, rental costs can be significantly lowered by using a few resources (sometimes, even just two) to complete the task. Thus, the computing task is switched from one resource to another to exploit the cloud provider’s price-performance schedule. Cloud computing has been recognized as an economically attractive computing environment whose adoption has been growing over time. However, providers (such as Amazon Web Services) offer a confusing and diverse set of computing resources with different configurations and unit rental costs. Our near-optimal solution is based on switching the computing task from one resource to another in way that leverages the relationship between the price and performance of the available computing resources. The performance of a given resource can vary randomly as well as be correlated with the performance of another (stronger or weaker) resource. We present a worst-case performance guarantee of the proposed solution. In addition, we study the performance using a detailed computational study and a real-world example of an actual company that can benefit from our proposed solution. In the computational study as well as the real-world example, the cost of our solution is usually about 15% – 25% lower than the benchmark solution of using the best single computing resource to process the computing task. Practicing IT managers can use our approach to migrate in-house solutions to the cloud in a cost-effective manner.