884 research outputs found

    Competitive-Ratio Approximation Schemes for Minimizing the Makespan in the Online-List Model

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    We consider online scheduling on multiple machines for jobs arriving one-by-one with the objective of minimizing the makespan. For any number of identical parallel or uniformly related machines, we provide a competitive-ratio approximation scheme that computes an online algorithm whose competitive ratio is arbitrarily close to the best possible competitive ratio. We also determine this value up to any desired accuracy. This is the first application of competitive-ratio approximation schemes in the online-list model. The result proves the applicability of the concept in different online models. We expect that it fosters further research on other online problems

    Scheduling in the Random-Order Model

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    Makespan minimization on identical machines is a fundamental problem in online scheduling. The goal is to assign a sequence of jobs to m identical parallel machines so as to minimize the maximum completion time of any job. Already in the 1960s, Graham showed that Greedy is (2-1/m)-competitive [Graham, 1966]. The best deterministic online algorithm currently known achieves a competitive ratio of 1.9201 [Fleischer and Wahl, 2000]. No deterministic online strategy can obtain a competitiveness smaller than 1.88 [Rudin III, 2001]. In this paper, we study online makespan minimization in the popular random-order model, where the jobs of a given input arrive as a random permutation. It is known that Greedy does not attain a competitive factor asymptotically smaller than 2 in this setting [Osborn and Torng, 2008]. We present the first improved performance guarantees. Specifically, we develop a deterministic online algorithm that achieves a competitive ratio of 1.8478. The result relies on a new analysis approach. We identify a set of properties that a random permutation of the input jobs satisfies with high probability. Then we conduct a worst-case analysis of our algorithm, for the respective class of permutations. The analysis implies that the stated competitiveness holds not only in expectation but with high probability. Moreover, it provides mathematical evidence that job sequences leading to higher performance ratios are extremely rare, pathological inputs. We complement the results by lower bounds for the random-order model. We show that no deterministic online algorithm can achieve a competitive ratio smaller than 4/3. Moreover, no deterministic online algorithm can attain a competitiveness smaller than 3/2 with high probability

    Scheduling in the Secretary Model

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    This paper studies online makespan minimization in the secretary model. Jobs, specified by their processing times, are presented in a uniformly random order. The input size n is known in advance. An online algorithm has to non-preemptively assign each job permanently and irrevocably to one of m parallel and identical machines such that the expected time it takes to process them all, the makespan, is minimized. We give two deterministic algorithms. First, a straightforward adaptation of the semi-online strategy Light Load [Albers and Hellwig, 2012] provides a very simple approach retaining its competitive ratio of 1.75. A new and sophisticated algorithm is 1.535-competitive. These competitive ratios are not only obtained in expectation but, in fact, for all but a very tiny fraction of job orders. Classically, online makespan minimization only considers the worst-case order. Here, no competitive ratio below 1.885 for deterministic algorithms and 1.581 using randomization is possible. The best randomized algorithm so far is 1.916-competitive. Our results show that classical worst-case orders are quite rare and pessimistic for many applications. We complement our results by providing first lower bounds. A competitive ratio obtained on nearly all possible job orders must be at least 1.257. This implies a lower bound of 1.043 for both deterministic and randomized algorithms in the general model

    Randomized algorithms for fully online multiprocessor scheduling with testing

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    We contribute the first randomized algorithm that is an integration of arbitrarily many deterministic algorithms for the fully online multiprocessor scheduling with testing problem. When there are two machines, we show that with two component algorithms its expected competitive ratio is already strictly smaller than the best proven deterministic competitive ratio lower bound. Such algorithmic results are rarely seen in the literature. Multiprocessor scheduling is one of the first combinatorial optimization problems that have received numerous studies. Recently, several research groups examined its testing variant, in which each job JjJ_j arrives with an upper bound uju_j on the processing time and a testing operation of length tjt_j; one can choose to execute JjJ_j for uju_j time, or to test JjJ_j for tjt_j time to obtain the exact processing time pjp_j followed by immediately executing the job for pjp_j time. Our target problem is the fully online version, in which the jobs arrive in sequence so that the testing decision needs to be made at the job arrival as well as the designated machine. We propose an expected (φ+3+1)(≈3.1490)(\sqrt{\varphi + 3} + 1) (\approx 3.1490)-competitive randomized algorithm as a non-uniform probability distribution over arbitrarily many deterministic algorithms, where φ=5+12\varphi = \frac {\sqrt{5} + 1}2 is the Golden ratio. When there are two machines, we show that our randomized algorithm based on two deterministic algorithms is already expected 3φ+313−7φ4(≈2.1839)\frac {3 \varphi + 3 \sqrt{13 - 7\varphi}}4 (\approx 2.1839)-competitive. Besides, we use Yao's principle to prove lower bounds of 1.66821.6682 and 1.65221.6522 on the expected competitive ratio for any randomized algorithm at the presence of at least three machines and only two machines, respectively, and prove a lower bound of 2.21172.2117 on the competitive ratio for any deterministic algorithm when there are only two machines.Comment: 21 pages with 1 plot; an extended abstract to be submitte

    Multi-processor Scheduling to Minimize Flow Time with epsilon Resource Augmentation

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    We investigate the problem of online scheduling of jobs to minimize flow time and stretch on m identical machines. We consider the case where the algorithm is given either (1 + ε)m machines or m machines of speed (1 + ε), for arbitrarily small ε \u3e 0. We show that simple randomized and deterministic load balancing algorithms, coupled with simple single machine scheduling strategies such as SRPT (shortest remaining processing time) and SJF (shortest job first), are O(poly(1/ε))-competitive for both flow time and stretch. These are the first results which prove constant factor competitive ratios for flow time or stretch with arbitrarily small resource augmentation. Both the randomized and the deterministic load balancing algorithms are non- migratory and do immediate dispatch of jobs. The randomized algorithm just allocates each incoming job to a random machine. Hence this algorithm is non- clairvoyant, and coupled with SETF (shortest elapsed time first), yields the first non-clairvoyant algorithm which is con- stant competitive for minimizing flow time with arbitrarily small resource augmentation. The deterministic algorithm that we analyze is due to Avrahami and Azar. For this algorithm, we show O(1/ε)-competitiveness for total flow time and stretch, and also for their Lp norms, for any fixed p ≥ 1

    Nash Social Welfare in Selfish and Online Load Balancing

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    In load balancing problems there is a set of clients, each wishing to select a resource from a set of permissible ones, in order to execute a certain task. Each resource has a latency function, which depends on its workload, and a client's cost is the completion time of her chosen resource. Two fundamental variants of load balancing problems are {\em selfish load balancing} (aka. {\em load balancing games}), where clients are non-cooperative selfish players aimed at minimizing their own cost solely, and {\em online load balancing}, where clients appear online and have to be irrevocably assigned to a resource without any knowledge about future requests. We revisit both selfish and online load balancing under the objective of minimizing the {\em Nash Social Welfare}, i.e., the geometric mean of the clients' costs. To the best of our knowledge, despite being a celebrated welfare estimator in many social contexts, the Nash Social Welfare has not been considered so far as a benchmarking quality measure in load balancing problems. We provide tight bounds on the price of anarchy of pure Nash equilibria and on the competitive ratio of the greedy algorithm under very general latency functions, including polynomial ones. For this particular class, we also prove that the greedy strategy is optimal as it matches the performance of any possible online algorithm

    Online Mixed Packing and Covering

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    Recent work has shown that the classical framework of solving optimization problems by obtaining a fractional solution to a linear program (LP) and rounding it to an integer solution can be extended to the online setting using primal-dual techniques. The success of this new framework for online optimization can be gauged from the fact that it has led to progress in several longstanding open questions. However, to the best of our knowledge, this framework has previously been applied to LPs containing only packing or only covering constraints, or minor variants of these. We extend this framework in a fundamental way by demonstrating that it can be used to solve mixed packing and covering LPs online, where packing constraints are given offline and covering constraints are received online. The objective is to minimize the maximum multiplicative factor by which any packing constraint is violated, while satisfying the covering constraints. Our results represent the first algorithm that obtains a polylogarithmic competitive ratio for solving mixed LPs online. We then consider two canonical examples of mixed LPs: unrelated machine scheduling with startup costs, and capacity constrained facility location. We use ideas generated from our result for mixed packing and covering to obtain polylogarithmic-competitive algorithms for these problems. We also give lower bounds to show that the competitive ratios of our algorithms are nearly tight
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