Weighted group search on a line & implications to the priority evacuation problem

We study two optimization problems in a multiprocessor environment in the presence of set-up costs. The first problem involves multiple parallel searchers (e.g., robots) that must locate a target which lies in one of many concurrent rays, and at an unknown position from their common origin. Every time a searcher turns direction, it incurs a turn cost. The objective is to derive a search strategy for locating the target as efficiently as possible. The second problem involves contract algorithms, namely algorithms in which the available computation time is specified prior to their execution. In particular, we seek a schedule of executions of contract algorithms for several different problems in identical parallel processors so as to efficiently simulate an interruptible algorithm, assuming that each execution incurs a given set-up cost. The performance of the search and scheduling strategies are evaluated by means of well-established measures, namely the competitive ratio and the acceleration ratio, respectively. In this paper we provide near-optimal strategies for the above problems, using an approach based on infinite linear-programming formulations. More precisely, we present a search strategy (resp. schedule) which is optimal when the number of rays (resp. problems) is a multiple of the number of searchers (resp. processors). For the general case, we show that the corresponding solutions are very close to optimal.

This paper concerns the mechanism design for online scheduling in a strategic setting. In this setting, each job is owned by a self-interested agent who may misreport the release time, deadline, length, and value of her job, while we need to determine not only the schedule of the jobs, but also the payment of each agent. We focus on the design of incentive compatible (IC) mechanisms, and study the maximization of social welfare (i.e., the aggregated value of completed jobs) by competitive analysis. We first derive two lower bounds on the competitive ratio of any deterministic IC mechanism to characterize the landscape of our research. We then propose a deterministic IC mechanism and show that such a simple mechanism works very well for both the preemption-restart model and the preemption-resume model. We show the mechanism can achieve the optimal competitive ratio of 5 for equal-length jobs and a near optimal competitive ratio (within a constant factor) for unequal-length jobs.

We consider the following non-preemptive semi-online scheduling problem. Jobs with non-increasing sizes arrive one by one to be scheduled on two uniformly related machines, with the goal of minimizing the makespan. We analyze both the optimal overall competitive ratio, and the optimal competitive ratio as a function of the speed ratio (q ≥ 1) between the two machines. We show that the greedy algorithm LPT has optimal competitive ratio 1/4(1 + √17) ≈ 1.28 overall, but does not have optimal competitive ratio for every value of q. We determine the intervals of q where LPT is an algorithm of optimal competitive ratio, and design different algorithms of optimal competitive ratio for the intervals where it fails to be the best algorithm. As a result, we give a tight analysis of the competitive ratio for every speed ratio.KeywordsInput SequenceCompetitive RatioSpeed RatioIdentical MachineOnline ScheduleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Online algorithms are a hallmark of worst case optimization under uncertainty. On the other hand, in practice, the input is often far from worst case, and has some predictable characteristics. A recent line of work has shown how to use machine learned predictions to circumvent strong lower bounds on competitive ratios in classic online problems such as ski rental and caching. We study how predictive techniques can be used to break through worst case barriers in online scheduling. The makespan minimization problem with restricted assignments is a classic problem in online scheduling theory. Worst case analysis of this problem gives Ω(log m) lower bounds on the competitive ratio in the online setting. We identify a robust quantity that can be predicted and then used to guide online algorithms to achieve better performance. Our predictions are compact in size, having dimension linear in the number of machines, and can be learned using standard off the shelf methods. The performance guarantees of our algorithms depend on the accuracy of the predictions, given predictions with error η, we show how to construct O(log η) competitive fractional assignments. We then give an online algorithm that rounds any fractional assignment into an integral schedule. Our algorithm is O((log log m)3)-competitive and we give a nearly matching Ω(log log m) lower bound for online rounding algorithms.1 Altogether, we give algorithms that, equipped with predictions with error η, achieve O(log η (log log m)3) competitive ratios, breaking the Ω(log m) lower bound even for moderately accurate predictions.

We consider the problem of on-line call admission and routing on trees and meshes. Previous work gave randomized on-line algorithms for these problems and proved that they have optimal (up to constant factors) competitive ratios. However, these algorithms can obtain very low profit with high probability. We investigate the question of devising for these problems on-line competitive algorithms that also guarantee a "good" solution with "good" probability. We give a new family of randomized algorithms with asymptotically optimal competitive ratios and "good" probability to get a profit close to the expectation. We complement these results by providing bounds on the probability of any optimally competitive randomized on-line algorithm for the problems we consider to get a profit close to the expectation. To the best of our knowledge, this is the first study of the relationship between the tail distribution and the competitive ratio of randomized on-line benefit algorithms.

The peak-demand charge motivates large-load customers to flatten their demand curves, while their self-owned renewable generations aggravate demand fluctuations. Thus, it is attractive to utilize energy storage for shaping real-time loads and reducing electricity bills. In this paper, we propose the first peak-aware competitive online algorithm for leveraging stored energy (e.g., in fuel cells) to minimize peak-demand charges. Our algorithm decides the discharging quantity slot by slot to maintain the optimal worst-case performance guarantee (namely, competitive ratio) among all deterministic online algorithms. Interestingly, we show that the best competitive ratio can be computed by solving a linear number of linear-fractional problems. We can also extend our competitive algorithm and analysis to improve the average-case performance and consider short-term prediction.

We study competitive online algorithms for EV (electrical vehicle) charging under the scenario of an aggregator serving a large number of EVs together with its background load, using both its own renewable energy (for free) and the energy procured from the external grid. The goal of the aggregator is to minimize its peak procurement from the grid, subject to the constraint that each EV has to be fully charged before its deadline. Further, the aggregator can predict the future demand and the renewable energy supply with some levels of uncertainty. The key challenge here is how to develop a model that captures the prior knowledge from such prediction, and how to best utilize this prior knowledge to reduce the peak under future uncertainty. In this paper, we first propose a 2-level increasing precision model (2-IPM), to capture the system uncertainty. We develop a powerful computation approach that can compute the optimal competitive ratio under 2-IPM over any online algorithm, and also online algorithms that can achieve the optimal competitive ratio. A dilemma for online algorithm design is that an online algorithm with good competitive ratio may exhibit poor average-case performance. We then propose a new Algorithm-Robustification procedure that can convert an online algorithm with reasonable average-case performance to one with both the optimal competitive ratio and good average-case performance. The robustified version of a well-known heuristic algorithm, Receding Horizon Control (RHC), is found to demonstrate superior performance via trace-based simulations.

We consider the problem of searching on a line using n mobile robots, of which at most f are faulty, and the remaining are reliable. The robots start at the same location and move in parallel along the line with the same speed. There is a target placed on the line at a location unknown to the robots. Reliable robots can find the target when they reach its location, but faulty robots cannot detect the target. Our goal is to design a parallel algorithm minimizing the competitive ratio, represented by the worst case ratio between the time of arrival of the first reliable robot at the target, and the distance from the source to the target. If $$n \ge 2f+2$$, there is a simple algorithm with a competitive ratio of 1. For $$f< n < 2f+2$$ we develop a new class of algorithms, called proportional schedule algorithms. For any given (n, f), we give a proportional schedule algorithm A(n, f), whose competitive ratio is $$\begin{aligned} \left( \frac{4f+4}{n} \right) ^{\frac{2f+2}{n}} \left( \frac{4f+4}{n} -2 \right) ^{1-\frac{2f+2}{n}} + 1. \end{aligned}$$Setting $$a=n/f$$ as a constant, the asymptotic competitive ratio is $$\left( 4/a \right) ^{2/a} \left( 4/a -2\right) ^{1-2/a } + 1$$. Our search algorithm is easily seen to be optimal for the case $$n=f+1$$. We also show that as n tends to $$\infty $$ the competitive ratio of our algorithm for the case $$n = 2f+1$$ approaches 3 and this is optimal. More precisely, we show that asymptotically (after excluding small order terms), the competitive ratio of our proportional schedule algorithm $$A(2f+1,f)$$ is at most $$3 + \frac{4\ln n}{n}$$, while any search algorithm has a lower bound $$3 + \frac{2\ln n}{n}$$ on its competitive ratio.

We introduce the compound interest rate into the continuous version of the online leasing problem and discuss the generalized model by competitive analysis. On the one hand, the optimal deterministic strategy and its competitive ratio are obtained; on the other hand, a nearly optimal randomized strategy is constructed and a lower bound for the randomized competitive ratios is proved by Yao′s principle. With the help of numerical examples, the theoretical results show that the interest rate puts off the purchase date and diminishes the uncertainty involved in the decision making.

This paper explores the scheduling of hydroelectric station during the flood season when the flood is hard to predict. Specifically, how to regulate the capacity of reservoir during the flood season to decrease the loss and maintain the generation efficiency from hydroelectric station will be elaborated in present work. In order to achieve this objective, online algorithm will be applied here to analyze optimal strategy, which manages the reservoir scheduling in terms of online strategy and competitive analysis. We give the online strategy LLS-algorithm with a competitive ratio c <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LLS</sub> , and analyze the properties of algorithms. Then the evidences from Ankang reservoir are collected to test the competitive ratio of online strategy. The LLS-algorithm effectively reduces the total loss of hydroelectric station during flood season. The model and results of the study have guiding significance and reference value for decision makers facing the similar situation.

We consider the problem of online scheduling a single machine with family setups under job availability. A setup must be scheduled when the next job comes from a different family from the last completed one, if any. The aim is to minimize the total completion time of all jobs. For the special case of identical processing times, we provide a lower bound for the competitive ratio and an online algorithm with its competitive analysis.

We integrate probability distribution into pure competitive analysis to improve the performance measure of competitive analysis, since input sequences of the leasing problem have simple structure and favorably statistical property. Let input structures be the characteristic of geometric distribution, and we obtain optimal on-line algorithms and their competitive ratios. Moreover, the introducing of interest rate would diminish the uncertainty involved in the process of decision making and put off the optimal purchasing date.

Let f be a function on a set of variables V. For each x e V, let c(x) be the cost of reading the value of x. An algorithm for evaluating f is a strategy for adaptively identifying and reading a set of variables U ⊆ V whose values uniquely determine the value of f. We are interested in finding algorithms which minimize the cost incurred to evaluate f in the above sense. Competitive analysis is employed to measure the performance of the algorithms. We study two variants of the above problem. First we consider the classical setting in which one assumes that the algorithm knows the cost c(x), for each x e V. For the case where f is a monotone boolean function which is representable by a threshold tree, we provide a polynomial time algorithm with the best possible competitive ratio γfc for each fixed cost function c(·). Remarkably, the best known result for the same class of functions is a pseudo-polynomial algorithm with competitiveness 2γfc. For the class of game tree functions our polynomial time algorithm attains, for each fixed cost function, the same competitiveness as the best known algorithm for the same class of functions, which instead runs in pseudo-polynomial time.In the second part of the paper, we study a novel variant of the problem. Here, we assume that the cost function is not known in advance and some preemption is allowed in the reading operations. This model has applications, e.g., when reading a variable coincides with obtaining the output of a job on a CPU and the cost is the CPU time. In such a case, it is reasonable to assume that no exact knowledge of the cost is available. We define a new algorithm for this problem based on the solution of a Linear Program. We show the optimality of our algorithm for the class of monotone boolean functions representable by AND-OR-trees. We also show a sub-optimal implementation for general monotone boolean functions.

The single machine semi-online scheduling problem with the objective ofminimizing total completion time is investigated with the assumption that theratio of the longest to the shortest processing time is not greater than aconstant $\gamma$. A semi-online algorithm is designed and its competitiveratio is proven to be $1+ \frac{\gamma - 1}{1 + \sqrt {1 + \gamma (\gamma -1)}}$. The competitive analysis method is as following: it starts from an arbitrary instance andmodifies the instance towards the possible structure of the worst-case instancewith respect to the given online algorithm. The modification guarantees thatthe performance ratio does not decrease. Eventually, it comes up with arelatively simple instance with a special structure, whose performance ratiocan be directly analyzed and serves as an upper bound on the competitive ratio.

Bin packing is a classic optimization problem with a wide range of applications, from load balancing to supply chain management. In this work, we study the online variant of the problem, in which a sequence of items of various sizes must be placed into a minimum number of bins of uniform capacity. The online algorithm is enhanced with a potentially erroneous prediction concerning the frequency of item sizes in the sequence. We design and analyze online algorithms with efficient tradeoffs between the consistency, which is the competitive ratio assuming no prediction error, and the robustness, which is the competitive ratio under adversarial error. Moreover, we demonstrate that the performance of our algorithm degrades near-optimally as a function of the prediction error. This is the first theoretical and experimental study of online bin packing under competitive analysis in the realistic setting of learnable predictions. Previous work addressed only extreme cases with respect to the prediction error and relied on overly powerful and error-free oracles.