Tight Competitive Ratio Research Articles

Online early work scheduling on parallel machines

We consider non-preemptive online parallel-machine scheduling with a common due date to maximize the total early work of all the jobs, i.e., the total processing time of the jobs (or parts) completed before the common due date. For the general case of m machines, we provide a parameter lower bound with respect to m. For the online algorithm, we first show that the tight competitive ratio of the classical list scheduling (LS) algorithm is 43. We then improve the upper bound on the competitive ratio for the previous algorithm, EFFm, to 1.2956. Additionally, we present a formula to compute the upper bound on the competitive ratio for any given m. For the case of three machines, we improve the lower bound to 1.1878 and propose an improved online algorithm with a tight competitive ratio of 1.2483.

Algorithms for maximum social welfare of online random trading

In an online random trading problem, m sellers and n buyers arrive in a random sequential order to meet a decision maker. Each seller possesses an item and each buyer demands an item. All items are identical. Each agent (seller or buyer) has a positive valuation on one item and reveals her valuation to the decision maker when she arrives. The decision maker, who knows only m and n in advance, uses an online algorithm to make an irrevocable decision on whether or not to trade with the arriving agent at her arrival time. We design online algorithms to maximize the social welfare, i.e., the expected total valuation of the agents who possess items on hand at the end of trading process. For the single-buyer trading, our algorithm achieves a tight competitive ratio of 1+1/m. For the single-seller trading, when n tends to infinity, we establish lower bound 3.258 and upper bound 4.189 on the competitive ratio. When both m and n are sufficiently large, our algorithm achieves an asymptotic competitive ratio no more than 1+O(m−1/3lnm).

Online Generalized Network Design Under (Dis)Economies of Scale

We consider a general online network design problem in which a sequence of N requests arrive over time, each of which needs to use some subset of the available resources E. The cost incurred by a resource [Formula: see text] is some function fe of the total load [Formula: see text] on that resource. The objective is to minimize the total cost [Formula: see text]. We focus on cost functions that exhibit (dis)economies of scale. These functions are of the form [Formula: see text] if x > 0 (and zero if x = 0), in which the exponent [Formula: see text]. Optimization problems under these functions have received significant recent attention because of applications in energy-efficient computing. Our main result is a deterministic online algorithm with tight competitive ratio [Formula: see text] when αe is constant for all [Formula: see text]. This framework is applicable to a variety of network design problems in undirected and directed graphs, including multicommodity routing, Steiner tree/forest connectivity, and set connectivity. In fact, our online competitive ratio even matches the previous best (offline) approximation ratio. We also demonstrate the practical performance of our online algorithm on instances of multicommodity routing. Funding: This work was supported by the Directorate for Computer and Information Science and Engineering [Grant CCF-1750127] and the Division of Civil, Mechanical and Manufacturing Innovation [Grant CMMI-1940766].

Online hierarchical parallel-machine scheduling in shared manufacturing to minimize the total completion time

We consider online hierarchical scheduling on m identical machines in shared manufacturing to minimize the total completion time. Each job has a unit-size processing time. The jobs arrive one by one and must be assigned to one of the m machines before the next job arrives. The jobs with a lower hierarchy can only be processed on the first k machines, and the jobs with a higher hierarchy can be processed on any one of the m machines. We first show that the lower bound of the problem is at least where Proposing a greedy algorithm, we show that its tight competitive ratio is by analyzing a set of instances that must contain a worst-case instance, which is different from the general method of calculating the competitive ratio. In addition, for the case where k = 1, we present an improved online algorithm with a tight competitive ratio of which is optimal for Numerical experiments show that the greedy online algorithm has good performance, especially when k approaches 1 or m.

Tight competitive analyses of online car-sharing problems

The online car-sharing problem finds many real-world applications. The problem, proposed by Luo, Erlebach and Xu in 2018, mainly focuses on an online model in which there are two locations: 0 and 1, and k total cars. Each request which specifies its pick-up time and pick-up location (among 0 and 1, and the other is the drop-off location) is released in each stage a fixed amount of time before its specified start (i.e. pick-up) time. The time between the booking (i.e. released) time and the start time is enough to move empty cars between 0 and 1 for relocation if they are not used in that stage. The model, called kS2L-F, assumes that requests in each stage arrive sequentially regardless of the same booking time and the decision (accept or reject) must be made immediately. The goal is to accept as many requests as possible. In spite of only two locations, the analysis does not seem easy and the (tight) competitive ratio (CR) is only known to be 2 for k=2 and 1.5 for a restricted value of k, i.e., a multiple of three. In this paper, we remove all the holes of unknown CR's; namely we prove that the CR is 2kk+⌊k/3⌋ for all k≥2. Furthermore, if the algorithm can delay its decision until all requests have come in each stage, the CR is improved to roughly 4/3. We can take this advantage even further; precisely we can achieve a CR of 2+R3 if the number of requests in each stage is at most Rk, 1≤R≤2, where we do not have to know the value of R in advance. Finally we demonstrate that randomization also helps to get (slightly) better CR's, and prove some lower bounds to show the tightness.

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.

Online Station Assignment for Electric Vehicle Battery Swapping

This paper investigates the online station assignment for (commercial) electric vehicles (EVs) that request battery swapping from a central operator, i.e., in the absence of future information a battery swapping service station has to be assigned instantly to each EV upon its request. Based on EVs’ locations, the availability of fully-charged batteries at service stations in the system, as well as traffic conditions, the assignment aims to minimize cost to EVs and congestion at service stations. Inspired by a polynomial-time offline solution via a bipartite matching approach, we develop an efficient and implementable online station assignment algorithm that provably achieves the tight (optimal) competitive ratio under mild conditions. Monte Carlo experiments on a real transportation network by Baidu Maps show that our algorithm performs reasonably well on realistic inputs, even with a certain amount of estimation error in parameters.

Non-Greedy Online Steiner Trees on Outerplanar Graphs

This paper addresses the classic online Steiner tree problem on edge-weighted graphs. It is known that a greedy (nearest neighbor) online algorithm has a tight competitive ratio for wide classes of graphs, such as trees, rings, any class including series-parallel graphs, and unweighted graphs with bounded diameter. However, we do not know any greedy or non-greedy tight deterministic algorithm for other classes of graphs. In this paper, we observe that a greedy algorithm is $$\Omega (\log n)$$ -competitive on outerplanar graphs, where n is the number of vertices, and propose a 5.828-competitive deterministic algorithm on outerplanar graphs. Our algorithm connects a requested vertex and the tree constructed thus far using a path that is constant times longer than the distance between them. We also present a lower bound of 4 for arbitrary deterministic online Steiner tree algorithms on outerplanar graphs.

Fully Online Matching

We introduce a fully online model of maximum cardinality matching in which all vertices arrive online. On the arrival of a vertex, its incident edges to previously arrived vertices are revealed. Each vertex has a deadline that is after all its neighbors’ arrivals. If a vertex remains unmatched until its deadline, then the algorithm must irrevocably either match it to an unmatched neighbor or leave it unmatched. The model generalizes the existing one-sided online model and is motivated by applications including ride-sharing platforms, real-estate agency, and so on. We show that the Ranking algorithm by Karp et al. (STOC 1990) is 0.5211-competitive in our fully online model for general graphs. Our analysis brings a novel charging mechanic into the randomized primal dual technique by Devanur et al. (SODA 2013), allowing a vertex other than the two endpoints of a matched edge to share the gain. To our knowledge, this is the first analysis of Ranking that beats 0.5 on general graphs in an online matching problem, a first step toward solving the open problem by Karp et al. (STOC 1990) about the optimality of Ranking on general graphs. If the graph is bipartite, then we show a tight competitive ratio ≈0.5671 of Ranking. Finally, we prove that the fully online model is strictly harder than the previous model as no online algorithm can be 0.6317 < 1- 1/e-competitive in our model, even for bipartite graphs.

The Mobile Server Problem

We introduce the Mobile Server problem, inspired by current trends to move computational tasks from cloud structures to multiple devices close to the end user. An example of this is embedded systems in autonomous cars that communicate to coordinate their actions. Our model is a variant of the classical Page Migration problem. More formally, we consider a mobile server holding a data page. The server can move in the Euclidean space (of arbitrary dimension). In every round, requests for data items from the page pop up at arbitrary points in the space. The requests are served, each at a cost of the distance from the requesting point and the server, and the mobile server may move, at a cost D times the distance traveled for some constant D . We assume a maximum distance m that the server is allowed to move per round. We show that no online algorithm can achieve a competitive ratio independent of the length of the input sequence in this setting. Hence, we augment the maximum movement distance of the online algorithms to (1+&delta) times the maximum distance of the offline solution. We provide a deterministic algorithm that is simple to describe and works for multiple variants of our problem. The algorithm achieves almost tight competitive ratios independent of the length of the input sequence. Our algorithm also achieves a constant competitive ratio without resource augmentation in a variant where the movement of clients is also restricted.

Online hole healing for sensor coverage

Many applications of wireless sensor networks require a quality of service including coverage and connectivity. However, both coverage and connectivity may be lost due to failure of sensors, in which case some holes appear. To heal holes, a hybrid wireless sensor system with mobile sensors has been proposed in the literature. Redundant mobile sensors can move to heal holes. Limited by energy supply, the largest distance that a mobile sensor can move cannot be too large. Such a consideration calls for cascade hole healing. This paper studies online cascade hole healing problem in which holes appear online and should be healed immediately without knowing future locations of holes. Two targets are considered, aiming to minimize the total energy consumption and the largest individual energy consumption, respectively. We show that naive greedy strategy does not work well for both targets, having exponentially large competitive ratios in a worst case. Then, we present two online algorithms with theoretically guaranteed tight competitive ratios. Extensive experiments show that our algorithm approximates offline optimal solutions fairly well.

Online packing of arbitrary sized items into designated and multipurpose bins

We consider an online multitype bin-packing problem with two item types. This setting can be motivated by transport applications in which some items may be shipped either in dry shipping containers or more costly refrigerated ones, while other items can only be transported in refrigerated containers. The problem was introduced by Goldberg and Karhi [Omega 71:85-92, 2017] who focused on the case of up to two item types, three bin types and only two item sizes. For this special case a tight (also known as optimal) absolute competitive ratio was shown of approximately 1.618. Here we consider the general problem of arbitrary item sizes and show a lower bound on the absolute competitive ratio of any online algorithm that is a function of the bin costs. This bound in the worst-case is approximately 1.781. We then extend the first-fit method to our problem setting and prove an absolute competitive ratio bound that is a function of the bin costs. In the worst case this upper bound is approximately 1.930. In addition an upper bound of 1.750 is established on the worst-case asymptotic (as the number of items grows large) competitive ratio.

Buyback problem with discrete concave valuation functions

We discuss an online discrete optimization problem called the buyback problem. In the literature of the buyback problem, the valuation function representing the total value of selected elements is given by a linear function. In this paper, we consider a generalization of the buyback problem using nonlinear valuation functions. We propose an online algorithm for the problem with discrete concave valuation functions, and show that it achieves the tight competitive ratio, i.e., the competitive ratio of the proposed algorithm is equal to the known lower bound for the problem.

Algorithms for Online Matching, Assortment, and Pricing with Tight Weight-Dependent Competitive Ratios

Resource Allocation and Pricing in the Absence of a Demand Forecast

Tight Weight-Dependent Competitive Ratios for Online Edge-Weighted Matching, with Application to Revenue Management

Motivated by the dynamic assortment offerings and item pricings occurring in e-commerce, we study a general problem of allocating finite inventories to heterogeneous customers arriving sequentially. We analyze this problem under the framework of competitive analysis, where the sequence of customers is unknown and does not necessarily follow any pattern. Previous work in this area, studying online matching, advertising, and assortment problems, has focused on the case where each item can only be sold at a single price, resulting in algorithms which achieve the best-possible competitive ratio of 1-1/e. In this paper, we extend all of these results to allow for items having multiple feasible prices. Our algorithms achieve the best-possible weight-dependent competitive ratios, which depend on the sets of feasible prices given in advance. Our algorithms are also simple and intuitive; they are based on constructing a class of universal ``value functions'' which integrate the selection of items and prices offered. Finally, we test our algorithms on the publicly-available hotel data set of Bodea et al. (2009), where there are multiple items (hotel rooms) each with multiple prices (fares at which the room could be sold). We find that applying our algorithms, as a ``hybrid'' with algorithms which attempt to forecast and learn the future transactions, results in the best performance.

Total completion time minimization in online hierarchical scheduling of unit-size jobs

This paper investigates an online hierarchical scheduling problem on m parallel identical machines. Our goal is to minimize the total completion time of all jobs. Each job has a unit processing time and a hierarchy. The job with a lower hierarchy can only be processed on the first machine and the job with a higher hierarchy can be processed on any one of m machines. We first show that the lower bound of this problem is at least $$1+\min \{\frac{1}{m}, \max \{\frac{2}{\lceil x\rceil +\frac{x}{\lceil x\rceil }+3}, \frac{2}{\lfloor x\rfloor +\frac{x}{\lfloor x\rfloor }+3}\}$$1+min{1m,max{2źxź+xźxź+3,2źxź+xźxź+3}, where $$x=\sqrt{2m+4}$$x=2m+4. We then present a greedy algorithm with tight competitive ratio of $$1+\frac{2(m-1)}{m(\sqrt{4m-3}+1)}$$1+2(m-1)m(4m-3+1). The competitive ratio is obtained in a way of analyzing the structure of the instance in the worst case, which is different from the most common method of competitive analysis. In particular, when $$m=2$$m=2, we propose an optimal online algorithm with competitive ratio of $$16$$16$$/$$/$$13$$13, which complements the previous result which provided an asymptotically optimal algorithm with competitive ratio of 1.1573 for the case where the number of jobs n is infinite, i.e., $$n\rightarrow \infty $$nźź.

Profitable Scheduling on Multiple Speed-Scalable Processors

We present a new online algorithm for profit-oriented scheduling on multiple speed-scalable processors and provide a tight analysis of the algorithm’s competitiveness. Our results generalize and improve upon work by Chan et al. [2010], which considers a single speed-scalable processor. Using significantly different techniques, we can not only extend their model to multiprocessors but also prove an enhanced and tight competitive ratio for our algorithm. In our scheduling problem, jobs arrive over time and are preemptable. They have different workloads, values, and deadlines. The scheduler may decide not to finish a job but instead to suffer a loss equaling the job’s value. However, to process a job’s workload until its deadline the scheduler must invest a certain amount of energy. The cost of a schedule is the sum of lost values and invested energy. In order to finish a job, the scheduler has to determine which processors to use and set their speeds accordingly. A processor’s energy consumption is power P α ( s ) integrated over time, where P α ( s )= s α is the power consumption when running at speed s . Since we consider the online variant of the problem, the scheduler has no knowledge about future jobs. This problem was introduced by Chan et al. [2010] for the case of a single processor. They presented an online algorithm that is α α + 2 e α-competitive. We provide an online algorithm for the case of multiple processors with an improved competitive ratio of α α .

Online algorithms for 1-space bounded 2-dimensional bin packing and square packing

In this paper, we study 1-space bounded 2-dimensional bin packing and square packing. A sequence of rectangular items (square items) arrive one by one, each item must be packed into a square bin of unit size on its arrival without any information about future items. When packing items, 90°-rotation is allowed. 1-space bounded means there is only one “active” bin. If the “active” bin cannot accommodate the coming item, it will be closed and a new bin will be opened. The objective is to minimize the total number of bins used for packing all items in the sequence.Our contributions are as follows: For 1-space bounded 2-dimensional bin packing, we propose an online packing algorithm with a tight competitive ratio of 5.06. A lower bound of 3.17 on the competitive ratio is proven. Moreover, we study 1-space bounded square packing, where each item is a square with side length no more than 1. A 4.3-competitive algorithm is achieved, and a lower bound of 2.94 on the competitive ratio is given. All these bounds surpass the previously best known results.

On the optimality of list scheduling for online uniform machines scheduling

This paper considers classical online scheduling problems on uniform machines. We show the tight competitive ratio of LS for any combinations of speeds of three machines. We prove that LS is optimal when s3 ≥ s2 ≥ s1 = 1 and \({s_3^2\geq s_2^2+s_2s_3+s_2}\), where s1, s2, s3 are the speeds of three machines. On the other hand, LS can not be optimal for all combinations of machine speeds, even restricted to the case of 1 = s1 = s2 < s3. For m ≥ 4 machines, LS remains optimal when one machine has very large speed, and the remaining machines have the same speed.

Tight Bounds for Delay-Sensitive Aggregation

Distributed Computing and Networking This article studies the fundamental trade-off between delay and communication cost in networks. We consider an online optimization problem where nodes are organized in a tree topology. The nodes seek to minimize the time until the root is informed about the changes of their states and to use as few transmissions as possible. We derive an upper bound on the competitive ratio of O(min (h, c)) where h is the tree's height, and c is the transmission cost per edge. Moreover, we prove that this upper bound is tight in the sense that any oblivious algorithm has a ratio of at least Omega(min (h, c)). For chain networks, we prove a tight competitive ratio of Theta(min (root h, c)). Furthermore, we introduce a model for value-sensitive aggregation, where the cost depends on the number of transmissions and the error at the root.