We investigate online scheduling with commitment for parallel identical machines. Our objective is to maximize the total processing time of accepted jobs. As soon as a job has been submitted, the commitment constraint forces us to decide immediately whether we accept or reject the job. Upon acceptance of a job, we must complete it before its deadline @math that satisfies @math , with @math and @math being the processing time and the submission time of the job, respectively while @math is the slack of the system. Since the hard case typically arises for near-tight deadlines, we consider @math . We use competitive analysis to evaluate our algorithms. Our first main contribution is a deterministic preemptive online algorithm with an almost tight competitive ratio on any number of machines. For a single machine, the competitive factor matches the optimal bound @math of the greedy acceptance policy. Then the competitive ratio improves with an increasing number of machines and approaches @math as the number of machines converges to infinity. This is an exponential improvement over the greedy acceptance policy for small @math . In the non-preemptive case, we present a deterministic algorithm on @math machines with a competitive ratio of @math . This matches the optimal bound of @math of the greedy acceptance policy for a single machine while it again guarantees an exponential improvement over the greedy acceptance policy for small @math and large @math . In addition, we determine an almost tight lower bound that approaches @math for large @math and small @math .
For parallel identical machines, DasGupta and Palis @cite_13 consider preemption without migration and suggest a simple greedy algorithm. They claim the same competitive factor as for the single machine problem with preemption. They also give a lower bound of @math . For the non-preemptive case, Kim and Chwa @cite_21 apply Goldwasser's algorithm @cite_9 @cite_17 to parallel identical machines. Lee @cite_20 considers a model without commitment and presents a non-preemptive algorithm that adds many newly submitted jobs to a queue and decides later whether to schedule or to reject them. He claims a competitive ratio of @math for small slack values. This competitive ratio is slightly worse than our competitive ratio for the same problem with commitment.
@cite_9 @cite_21 @cite_13 @cite_20 @cite_17
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In this paper we provide a tight bound for the price of preemption for scheduling jobs on a single machine (or multiple machines). The input consists of a set of jobs to be scheduled and of an integer parameter @math . Each job has a release time, deadline, length (also called processing time) and value associated with it. The goal is to feasibly schedule a subset of the jobs so that their total value is maximal; while preemption of a job is permitted, a job may be preempted no more than k times. The price of preemption is the worst possible (i.e., largest) ratio of the optimal non-bounded-preemptive scheduling to the optimal k-bounded-preemptive scheduling. Our results show that allowing at most k preemptions suffices to guarantee a Θ( +1 n, log_k+1 P ) @math . As part of the proof, we introduce the notion of the Bounded-Degree Ancestor-Free Sub-Forest (BAS). We investigate the problem of computing the maximal-value BAS of a given forest and give a tight bound for the loss factor, which is Θ(log_k+1 n)$ as well, where n is the size of the original forest and k is the bound on the degree of the sub-forest. We study the following online preemptive scheduling problem: given a set of jobs with release times, deadlines, processing times and weights, schedule them so as to maximize the total value obtained. Unlike traditional scheduling problems, partially completed jobs can get partial values proportional to their amounts processed. Recently gave improved lower and upper bounds (1.236, 1.8) on the competitive ratio for this problem, the upper bound being achieved by using timesharing to simulate two equal-speed processors. In this paper we (1) give a new algorithm MIXED-k with competitive ratio 1 (1 − (k (k + 1)) k ) which approaches e (e−1) ≈ 1.582 when k →∞ , by using timesharing to simulate k equal-speed processors; (2) give an equivalent but much more practical algorithm MIX, which is e (e − 1)-competitive (independent of k), by timesharing the processor with different speeds (depending on the job weights), and use its interesting properties to devise an efficient implementation; (3) improve the lower bound to 1.25 by showing an identical lower bound for randomized algorithms; and (4) prove a lower bound of 1.618 on the competitive ratio when timesharing is not allowed, thus answering an open problem raised by Chang and Yap, showing that timesharing provably helps in giving better algorithms for this problem. The problem considered here is the same as the one discussed in [G. Galambos and G. J. Woeginger, eds., SIAM J. Comput., 22 (1993), pp. 349--355]. It is an m-machine online scheduling problem in which we wish to minimize the competitive ratio for the makespan objective. In this paper, we show that @math is a lower bound on this competitive ratio for m=4. In particular, we show how to force a lower bound of @math for any positive @math . This reduces the gap between the performance of known algorithms [S. Albers, in Proceedings of the 29th Annual ACM Symposium on Theory of Computing, ACM, New York, 1997, pp. 130--139] and the lower bound. The method used introduces an approach to building the task master's strategy. We study the competitiveness of online deadline scheduling problems. It is assumed that jobs are non-preemptive and we want to maximize, in an online manner, the sum of the length of jobs completed before their deadlines. When there is a single machine, Goldwasser [4] showed that the optimal deterministic competitiveness of this problem is 2+1 k, where each job of length L can be delayed for at least k • L before it is started, while still meeting its deadline. We consider the case that k < 1 and present an O((log 1 k ))-competitive randomized algorithm not only for a single machine but also for m machines where m = 1,2,•••, O(( log 1 k )).Of particular interest is our technique: we mainly consider deterministic algorithms for multiple machines in order to improve the randomized competitiveness for a single (or more) machine. Though this approach is not completely new, it is rather complicated in our case to design a deterministic algorithm for multiple machines. Specifically, we present an [m+1+ m • (1 k)(1 m)]-competitive deterministic algorithm, where m (≥ 2) machines are available to both online algorithms and the adversary.Finally we also study a related problem and present an improved algorithm.