Competitive Online Algorithm Research Articles

Technical Note—Near-Optimal Bayesian Online Assortment of Reusable Resources

Near-Optimal Bayesian Online Assortment of Reusable Resources Motivated by rental services in e-commerce, we consider revenue maximization in the online assortment of reusable resources for different types of arriving consumers. We design competitive online algorithms compared with the optimal online policy in the Bayesian setting, where consumer types are drawn independently from known heterogeneous distributions over time. In scenarios with large initial inventories, our main result is a near-optimal competitive algorithm for reusable resources. Our algorithm relies on an expected linear programming (LP) benchmark, solves this LP, and simulates the solution through independent randomized rounding. The main challenge is achieving inventory feasibility efficiently using these simulation-based algorithms. To address this, we design discarding policies for each resource, balancing inventory feasibility and revenue loss. Discarding a unit of a resource impacts future consumption of other resources, so we introduce postprocessing assortment procedures to design and analyze our discarding policies. Additionally, we present an improved competitive algorithm for nonreusable resources and evaluate our algorithms using numerical simulations on synthetic data.

An online joint replenishment problem combined with single machine scheduling

This paper considers a combination of the joint replenishment problem with single machine scheduling. There is a single resource, which is required by all the unit-time jobs, and a job can be started at time point t on the machine if and only if the machine does not process another job at t, and the resource is replenished between its release date and t. Each replenishment has a cost, which is independent of the amount replenished. The objective is to minimize the total replenishment cost plus the maximum flow time of the jobs. We consider the online variant of the problem, where the jobs are released over time, and once a job is inserted into the schedule, its starting time cannot be changed. We propose a deterministic 2-competitive online algorithm for the general input. Moreover, we show that for a certain class of inputs (so-called p-bounded input), the competitive ratio of the algorithm tends to sqrt{2} as the number of jobs tends to infinity. We also derive several lower bounds for the best competitive ratio of any deterministic online algorithm under various assumptions.

Two-Stage Stochastic Matching and Pricing with Applications to Ride Hailing

Two-Stage Matching and Pricing in Ride-Hailing Platforms Matching and pricing are two critical levers in two-sided marketplaces to connect demand and supply. The platform can produce more efficient matching and pricing decisions by batching the demand requests. We initiate the study of the two-stage stochastic matching problem with or without pricing to enable the platform to make improved decisions in a batch with an eye toward the imminent future demand requests. This problem is motivated in part by applications in online marketplaces, such as ride-hailing platforms. We design online competitive algorithms for vertex-weighted (or unweighted) two-stage stochastic matching for maximizing supply efficiency and two-stage joint matching and pricing for maximizing market efficiency. Using various techniques, such as introducing convex programming–based matching and graph decompositions, submodular maximization, and factor-revealing linear programs, we obtain either optimal competitive or improved approximation algorithms compared with naïve solutions. We enrich our theoretical study by data-driven numerical simulations using DiDi’s ride-sharing data sets.

MUTAA: An online trajectory optimization and task scheduling for UAV-aided edge computing

A novel UAV-aided edge computing system is proposed in this work, where UAV-aided edge nodes are dispatched to provide communication and computation assistance for completing tasks generated by ground clients (GCs). We formulate a trajectory design and task allocation problem (TDTAP), aiming at maximizing the sum of completed tasks of GCs by optimizing the proper trajectory for each UAV and scheduling tasks from each GC. It is impossible to solve the TDTAP problem directly in polynomial time since UAVs lack all GCs’ information, e.g., position and amount of tasks. To this end, we put forward an online iterative algorithm named Maximum UAV trajectory and Task Allocation Algorithm (MUTAA) to solve the TDTAP problem by jointly optimizing UAVs’ trajectory and GCs’ task scheduling. Unlike existing algorithms, MUTAA can make real-time decisions for each UAV without acquiring information from all GCs in advance. During each iteration, MUTAA consists of two sub-algorithms: (1) trajectory design algorithm TDA and (2) task allocation strategy TAS. Specifically, the preschedule step is used in TDA to find the proper trajectory for UAVs, and a competitive online algorithm, TAS, is proposed to schedule GCs’ tasks. Theoretical analysis proves that TAS is e/(e−1)-competitive, that is, it processes (e−1)/e (approximately 63%) tasks when compared with the optimal offline solution. Experimental results demonstrate that MUTAA completes 83% data on an average of the optimal offline solution, illustrating that the proposed algorithm MUTAA can be widely used in time-sensitive scenarios.

An Online Scheduling Problem on a Drop-Line Parallel Batch Machine with Delivery Times and Limited Restart

This paper studies online scheduling on an unbounded drop-line parallel batch machine with delivery times and limited restarts. A drop-line parallel batch machine is a system that can process some jobs simultaneously as a batch. Jobs in a batch have the same starting time and the completion time of a job is equal to its starting time plus its processing time. Limited restarts mean that a running batch containing at least one restarted job cannot be restarted again. The objective is to minimize the time by which all jobs have been delivered. We prove that any online algorithm has a competitive ratio of at least [Formula: see text], where [Formula: see text] is the positive solution of the equation [Formula: see text]. We provide a best possible [Formula: see text]-competitive online algorithm for the problem. Furthermore, we study the restricted problem with small delivery times and provide a best possible online algorithm with a competitive ratio of [Formula: see text].

Massive IoT Access With NOMA in 5G Networks and Beyond Using Online Competitiveness and Learning

This paper studies the problem of online user grouping, scheduling and power allocation in beyond 5G cellular-based Internet of things networks. Due to the massive number of devices trying to be granted to the network, non-orthogonal multiple access method is adopted in order to accommodate multiple devices in the same radio resource block. Different from most previous works, the objective is to maximize the number of served devices while allocating their transmission powers such that their real-time requirements as well as their limited operating energy are respected. First, we formulate the general problem as a mixed integer non-linear program (MINLP) that can be transformed easily to MILP for some special cases. Second, we study its computational complexity by characterizing the NP-hardness of different special cases. Then, by dividing the problem into multiple NOMA grouping and scheduling subproblems, efficient online competitive algorithms are proposed. Further, we show how to use these online algorithms and combine their solutions in a reinforcement learning setting to obtain the power allocation and hence the global solution to the problem. Our analysis are supplemented by simulation results to illustrate the performance of the proposed algorithms with comparison to optimal and state-of-the-art methods.

Competitive Online Convex Optimization With Switching Costs and Ramp Constraints

We investigate competitive online algorithms for online convex optimization (OCO) problems with linear in-stage costs, switching costs and ramp constraints. While OCO problems have been extensively studied in the literature, there are limited results on the corresponding online solutions that can attain small competitive ratios. We first develop a powerful computational framework that can compute an optimized competitive ratio based on the class of affine policies. Our computational framework can handle a fairly general class of costs and constraints. Compared with other competitive results in the literature, a key feature of our proposed approach is that it can handle scenarios where infeasibility may arise due to hard feasibility constraints. Second, we design a robustification procedure to produce an online algorithm that can attain good performance for both average-case and worst-case inputs. We conduct a case study on Network Functions Virtualization (NFV) orchestration and scaling to demonstrate the effectiveness of our proposed methods.

Competitive Algorithms for the Online Minimum Peak Appointment Scheduling

This paper describes a fundamental online scheduling problem called the minimum peak appointment scheduling (MPAS) problem. In this problem, there is a sequence of arriving appointments, each with specified required duration. The goal is to schedule the appointments in an online manner and to minimize the resulting peak utilization throughout the scheduling interval. While described as appointment scheduling, the MPAS has far broader applicability that captures other settings. The paper provides a formal modeling framework to analyze the aforementioned MPAS problem. It is shown that the offline version of the MPAS problem is identical to the offline version of the well-known bin packing problem. However, in the online variant, there is additional flexibility regarding how appointments are packed into chairs (bins). The paper describes the first competitive online algorithm to the MPAS problem called the harmonic rematching (HR) algorithm. The analysis shows that the HR algorithm has an asymptotic competitive ratio of 1.5. Considering that the current best lower bound on randomized online algorithms is 1.536, this highlights the fact that the HR algorithm performs better than any bin packing online algorithm in this setting.

Multi-Resource VNF Deployment in a Heterogeneous Cloud

The emerging paradigm of Network Function Virtualization (NFV) promises to shorten the renewal cycles of network functions and reduce the capital expenses by flexibly deploying virtualized network functions (VNFs) implementation on commodity servers. However, the required resource of each type (CPU, memory, etc.) for the running VNF should be provisioned to guarantee the performance when processing packets. This comes with different deployment cost, especially in a heterogeneous cloud consisting of a large number of network function platforms from various vendors. To optimally operate VNFs, it is necessary for the network operator to dynamically deploy VNFs in the expensive cloud infrastructures. In this article, we initiate the study of minimizing the deployment cost under multi-resource constraints in a heterogeneous cloud. We formulate multi-resource VNF deployment problem (MVDP) as an optimization program and prove its hardness. We propose an offline <inline-formula><tex-math notation="LaTeX">$(1, d+1)$</tex-math></inline-formula> -bicriteria approximation algorithm and an <inline-formula><tex-math notation="LaTeX">$(\mathcal {O}(1), \mathcal {O}(n \cdot \log n))$</tex-math></inline-formula> -competitive online algorithm to deploy VNFs in a scalable manner, where <inline-formula><tex-math notation="LaTeX">$d$</tex-math></inline-formula> is the number of resource types and <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> is the number of required VNFs. Large-scale simulations and DPDK-based OpenNetVM implementation show that our algorithms can reduce the overall cost by 34% and improve the performance in terms of multi-resource allocation.

Interval Job Scheduling With Machine Launch Cost

We study an interval job scheduling problem in distributed systems. We are given a set of interval jobs, with each job specified by a size, an arrival time and a processing length. Once a job arrives, it must be placed on a machine immediately and run for a period of its processing length without interruption. The homogeneous machines to run jobs have the same capacity limits such that at any time, the total size of the jobs running on any machine cannot exceed its capacity. Launching each machine incurs a fixed cost. After launch, a machine is charged a constant cost per time unit until it is terminated. The problem targets to minimize the total cost incurred by the machines for processing the given set of interval jobs. We focus on the algorithmic aspects of the problem in this article. For the special case where all the jobs have a unit size equal to the machine capacity, we propose an optimal offline algorithm and an optimal 2-competitive online algorithm. For the general case where jobs can have arbitrary sizes, we establish a non-trivial lower bound on the optimal solution. Based on this lower bound, we propose a 5-approximation algorithm in the offline setting. In the non-clairvoyant online setting, we design a $O(\mu)$ O ( μ ) -competitive Modified First-Fit algorithm which is near optimal ( $\mu$ μ is the max/min job processing length ratio). In the clairvoyant online setting, we propose an asymptotically optimal $O(\sqrt{\log \mu })$ O ( log μ ) -competitive algorithm based on our Modified First-Fit strategy.

Applying “Peeling Onion” approach for competitive analysis in online scheduling with rejection

In this study, we consider the classical online scheduling problem with rejection on identical parallel machines. In particular, a series of independent jobs arriving online over time will be scheduled on the identical machines with the flexibility of rejection, which implies that each job is either rejected with a penalty cost or accepted and scheduled on one of the identical machines. In addition, the knowledge of each job Jj, including the processing time pj, release date rj, and penalty cost ej, is disclosed upon arrival of this job. Our goal is to minimize the total completion time of the accepted jobs plus the total penalty cost of the rejected jobs. For this problem, we provide a deterministic polynomial time online algorithm named as Delayed Shortest Processing Time with Rejection, shortly as DSPTR. For the sake of analyzing the competitive ratio of DSPTR, we introduce an interesting and efficient approach entitled as “Peeling Onion”. By means of “Peeling Onion”, we demonstrate that DSPTR is a 2-competitive online algorithm and the bound is tight. We believe that the introduction of the analysis approach “Peeling Onion” could also be applied to other online optimization problems and yield a new perspective on competitive analysis in general.

Online machine minimization with lookahead

This paper studies the online machine minimization problem, where the jobs have real release times, uniform processing times and a common deadline. We investigate how the lookahead ability improves the performance of online algorithms. Two lookahead models are studied, that is, the additive lookahead and the multiplicative lookahead. At any time t, the online algorithm knows all the jobs to be released before time $$t+L$$ (or $$\beta \cdot t$$ ) in the additive (or multiplicative) lookahead model. We propose a $$\frac{e}{\alpha (e-1)+1}$$ -competitive online algorithm with the additive lookahead, where $$\alpha = \frac{L}{T} \le 1$$ and T is the common deadline of the jobs. For the multiplicative lookahead, we provide an online algorithm with a competitive ratio of $$\frac{\beta e}{(\beta -1) e +1}$$ , where $$\beta \ge 1$$ . Lower bounds are also provided for both of the two models, which show that our algorithms are optimal for two extreme cases, that is, $$\alpha = 0$$ (or $$\beta = 1$$ ) and $$\alpha = 1$$ (or $$\beta \rightarrow \infty $$ ), and remain a small gap for the cases in between. Particularly, for $$\alpha = 0$$ (or $$\beta = 1$$ ), the competitive ratio is e, which corresponds to the problem without lookahead. For $$\alpha = 1$$ (or $$\beta \rightarrow \infty $$ ), the competitive ratio is 1, which corresponds to the offline version (with full information).

Two-stage Matching and Pricing with Applications to Ride Hailing

Matching and pricing are two critical levers in two-sided marketplaces to connect demand and supply. The platform can produce more efficient matching and pricing decisions by batching the demand requests. We initiate the study of the two-stage stochastic matching problem, with or without pricing, to enable the platform to make improved decisions in a batch with an eye toward the imminent future demand requests. This problem is motivated in part by applications in online marketplaces such as ride hailing platforms. We design online competitive algorithms for vertex-weighted (or unweighted) two-stage stochastic matching for maximizing supply efficiency, and two-stage joint matching and pricing for maximizing market efficiency. In the former problem, using a randomized primal-dual algorithm applied to a family of balancing convex programs, we obtain the optimal 3/4 competitive ratio against the optimum offline benchmark. Using a factor revealing program and connections to submodular optimization, we improve this ratio against the optimum online benchmark to (1−1/e+1/e^2) ≈ 0.767 for the unweighted and 0.761 for the weighted case. In the latter problem, we design optimal 1/2-competitive joint pricing and matching algorithm by borrowing ideas from the ex-ante prophet inequality literature. We also show an improved (1−1/e)-competitive algorithm for the special case of demand efficiency objective using the correlation gap of submodular functions. Finally, we complement our theoretical study by using DiDi’s ride-sharing dataset for Chengdu city and numerically evaluating the performance of our proposed algorithms in practical instances of this problem.

Online traveling salesman problem with time cost and non-zealous server

Considering that the time of meeting the demands is very important for emergency vehicle and emergency vehicle can’t reject any request, we introduce a more realistic cost form into online traveling salesman problem(OL-TSP). When a new request arrives, if the salesman can’t serve the request immediately, per-unit-time cost will be generated. The goal is to minimize server’s total costs(travel makespan plus the per-unit-time costs). We consider the server is a non-zealous server and show that neither deterministic nor randomized online algorithms can achieve constant competitive ratio for OL-TSP on general metric space. While on truncated line segment and uniform metric space, we prove lower bounds, and present competitive online algorithms. Especially for the case with uniform metric space, we prove an optimal Greedy algorithm.

A Competitive Online Algorithm for Minimizing Total Weighted Completion Time on Uniform Machines

We consider the classic online scheduling problem on m uniform machines in the online setting where jobs arrive over time. Preemption is not allowed. The objective is to minimize total weighted completion time. An online algorithm based on the directly waiting strategy is proposed. Its competitive performance is proved to be max2smax1−1/2∑si,2smax/1+smax2.5−1/2m by the idea of instance reduction, where sm is the fastest machine speed after being normalized by the slowest machine speed.

Near-Optimal Bayesian Online Assortment of Reusable Resources

Motivated by the applications of rental services in e-commerce, we consider revenue maximization in online assortment of reusable resources for a stream of arriving consumers with different types. We design competitive online algorithms with respect to the optimum online policy in the Bayesian setting, in which types are drawn independently from known heterogeneous distributions over time. In the regime where the minimum of initial inventories c_min is large, our main result is a near-optimal 1-min(1/2,\sqrt{log(c_min)/c_min}) competitive algorithm for the general case of reusable resources. For the special case of non-reusable resources, we further show an improved near-optimal 1-1/\sqrt{c_min+3} competitive algorithm. Our algorithms rely on an expected LP benchmark for the problem, solve this LP, and simulate the solution through an independent randomized rounding. To obtain point-wise inventory feasibility in a computationally efficient fashion from these simulation-based algorithms, we use several technical ingredients to design -- one for each resource. These policies handle the trade-off between the inventory feasibility under reusability and the revenue loss of each of the resources. We also introduce post-processing assortment procedures that help with designing and analyzing our discarding policies, which might be of independent interest. We finally evaluate the performance of our algorithms using the simulation on synthetic data.

Online Scheduling of Task Graphs on Heterogeneous Platforms

Modern computing platforms commonly include accelerators. We target the problem of scheduling applications modeled as task graphs on hybrid platforms made of two types of resources, such as CPUs and GPUs. We consider that task graphs are uncovered dynamically, and that the scheduler has information only on the available tasks, i.e., tasks whose predecessors have all been completed. Each task can be processed by either a CPU or a GPU, and the corresponding processing times are known. Our study extends a previous $4\sqrt{m/k}$ 4 m / k -competitive online algorithm by Amaris et al. [1] , where $m$ m is the number of CPUs and $k$ k the number of GPUs ( $m\geq k$ m ≥ k ). We prove that no online algorithm can have a competitive ratio smaller than $\sqrt{m/k}$ m / k . We also study how adding flexibility on task processing, such as task migration or spoliation, or increasing the knowledge of the scheduler by providing it with information on the task graph, influences the lower bound. We provide a $(2\sqrt{m/k}+1)$ ( 2 m / k + 1 ) -competitive algorithm as well as a tunable combination of a system-oriented heuristic and a competitive algorithm; this combination performs well in practice and has a competitive ratio in $\Theta (\sqrt{m/k})$ Θ ( m / k ) . We also adapt all our results to the case of multiple types of processors. Finally, simulations on different sets of task graphs illustrate how the instance properties impact the performance of the studied algorithms and show that our proposed tunable algorithm performs the best among the online algorithms in almost all cases and has even performance close to an offline algorithm.

Hiring Secretaries over Time: The Benefit of Concurrent Employment

We consider a stochastic online problem where n applicants arrive over time, one per time step. Upon the arrival of each applicant, their cost per time step is revealed, and we have to fix the duration of employment, starting immediately. This decision is irrevocable; that is, we can neither extend a contract nor dismiss a candidate once hired. In every time step, at least one candidate needs to be under contract, and our goal is to minimize the total hiring cost, which is the sum of the applicants’ costs multiplied with their respective employment durations. We provide a competitive online algorithm for the case that the applicants’ costs are drawn independently from a known distribution. Specifically, the algorithm achieves a competitive ratio of 2.965 for the case of uniform distributions. For this case, we give an analytical lower bound of 2 and a computational lower bound of 2.148. We then adapt our algorithm to stay competitive even in settings with one or more of the following restrictions: (i) at most two applicants can be hired concurrently; (ii) the distribution of the applicants’ costs is unknown; (iii) the total number n of time steps is unknown. On the other hand, we show that concurrent employment is a necessary feature of competitive algorithms by proving that no algorithm has a competitive ratio better than [Formula: see text] if concurrent employment is forbidden.

Group parking permit problems

In this paper we study some generalizations of the parking permit problem (Meyerson, FOCS’05), in which we are given a demand rt∈{0,1} for instant of time t=0,…,T−1, along with K permit types with lengths of time δ1,…,δK and sub-additive costs. A permit is a pair (k,tˆ)∈[K]×Z+, and it covers interval [tˆ,tˆ+δk). We wish to find a minimum-cost set of permits that covers every t with rt=1. Meyerson gave deterministic O(K)-competitive and randomized O(lgK)-competitive online algorithms for this problem, as well as matching lower bounds.The first variant we propose is the multi parking permit problem, in which an arbitrary demand is given at each instant (rt∈Z+) and we may buy multiple permits to serve it. We prove that the offline version of this problem can be solved in polynomial time, and we show how to reduce it to the parking permit problem, while losing a constant cost factor. This approximation-preserving reduction yields a deterministic O(K)-competitive online algorithm and a randomized O(lgK)-competitive online algorithm. For a leasing variant of the Steiner network problem, these results imply a O(lgn)-approximation algorithm and a O(lgKlg|V|)-competitive online algorithm, where n is the number of requests and |V| is the size of the input metric.The second variant we propose is the group parking permit problem, in which we also have an arbitrary demand for each instant, and each permit of type k can be either a single permit, costing γk and covering one demand per instant of time, or a group permit, which costs M⋅γk for some constant M≥1 and covers an arbitrary number of demands in the interval covered by this permit. (I.e., group permits have infinite capacity.) For this version of the problem, we give an 8-approximation algorithm and a deterministic O(K)-competitive online algorithm. The first result yields an improvement on the previous best approximation algorithm for the leasing version of the rent-or-buy problem.Finally, we study the 2D parking permit problem, proposed by Hu, Ludwig, Richa and Schmid (2015), in which a permit type is defined by a length of time and an integer capacity. They presented a constant approximation algorithm and a deterministic O(K)-competitive online algorithm for a hierarchical version of the problem, but those algorithms have pseudo-polynomial running time. We show how to turn their algorithms into polynomial time algorithms. Moreover, these results yield approximation and competitive online algorithms for a hierarchical leasing version of the buy-at-bulk network design problem. We also show that their original pseudo-polynomial offline algorithm works for a more general version of the 2D parking permit problem, which we prove to be NP-hard.

An O (log k )-Competitive Algorithm for Generalized Caching

In the generalized caching problem, we have a set of pages and a cache of size k . Each page p has a size w p ≥ 1 and fetching cost c p for loading the page into the cache. At any point in time, the sum of the sizes of the pages stored in the cache cannot exceed k . The input consists of a sequence of page requests. If a page is not present in the cache at the time it is requested, it has to be loaded into the cache, incurring a cost of c p . We give a randomized O (log k )-competitive online algorithm for the generalized caching problem, improving the previous bound of O (log 2 k ) by Bansal, Buchbinder, and Naor (STOC’08). This improved bound is tight and of the same order as the known bounds for the classic paging problem with uniform weights and sizes. We use the same LP-based techniques as Bansal et al. but provide improved and slightly simplified methods for rounding fractional solutions online.