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Online scheduling is one of the most basic online problems. We are given a sequence of jobs in an online fashion, and they must be assigned to one of m ( ≥ 1 ) equivalent processors. Each job j with the processing time p ( j ) is given at the release time r ( j ) . An algorithm decides which processor is used to process j and must schedule it for time p ( j ) after time r ( j ) on the chosen processor. Many criteria have been proposed to evaluate the performance of scheduling algorithms, and Muthukrishnan et al. (FOCS 1999 and SIAM Journal on Computing 34 (2), 2005) used the stretch of a job j in an objective function, which is defined as the ratio of the amount of time which j spends in the system to its processing time p ( j ) . The cost of an algorithm is the total stretch to schedule all the given jobs, and our goal is to minimize it. Muthukrishnan et al. considered the case in which preemption is allowed, that is, any job can be stopped during the process and after a while the process can resume. In this paper, we show how to construct instances to obtain various lower bounds on the competitive ratio for each m . In the instances, m jobs are given regularly for a sufficiently large time span after a maliciously chosen time. The processing times of the given jobs are taken depending on the remaining processing times of uncompleted jobs at the start time of the “burst” of jobs. We prove that for the instances, the stretch of a job completed before the burst hardly affects the evaluation of a competitive ratio. Further, we provide a job sequence given before the burst for each m . Then, we can improve the previous lower bounds for any deterministic online algorithm for each m using the instances. For example, we obtain lower bounds of 1.228, 1.257 and 21 / 17 ≈ 1.235 for m = 1 , 2 and ∞, respectively. Moreover, we obtain the first non-trivial lower bounds for any randomized online algorithm for all m . [ABSTRACT FROM AUTHOR]

We consider a variant of the online buffer management problem in network switches, called the k -frame throughput maximization problem ( k -FTM). This problem models the situation where a large frame is fragmented into k packets and transmitted through the Internet, and the receiver can reconstruct the frame only if he/she accepts all the k packets. Kesselman et al. introduced this problem and showed that its competitive ratio is unbounded even when k = 2 . They also introduced an “order-respecting” variant of k -FTM, called k -OFTM, where inputs are restricted in some natural way. They proposed an online algorithm and showed that its competitive ratio is at most 2 k B ⌊ B / k ⌋ + k for any B ≥ k , where B is the size of the buffer. They also gave a lower bound of B ⌊ 2 B / k ⌋ for deterministic online algorithms when 2 B ≥ k and k is a power of 2. In this paper, we improve upper and lower bounds on the competitive ratio of k -OFTM. Our main result is to improve an upper bound of O ( k 2 ) by Kesselman et al. to 5 B + ⌊ B / k ⌋ − 4 ⌊ B / 2 k ⌋ = O ( k ) for B ≥ 2 k . Note that this upper bound is tight up to a multiplicative constant factor since the lower bound given by Kesselman et al. is Ω ( k ) . We also give two lower bounds. First we give a lower bound of 2 B ⌊ B / ( k − 1 ) ⌋ + 1 on the competitive ratio of deterministic online algorithms for any k ≥ 2 and any B ≥ k − 1 , which improves the previous lower bound of B ⌊ 2 B / k ⌋ by a factor of almost four. Next, we present the first nontrivial lower bound on the competitive ratio of randomized algorithms. Specifically, we give a lower bound of k − 1 against an oblivious adversary for any k ≥ 3 and any B . Since a deterministic algorithm, as mentioned above, achieves an upper bound of about 10 k , this indicates that randomization does not help too much. [ABSTRACT FROM AUTHOR]

We consider the one-dimensional version of the online unit clustering problem, proposed by Chan and Zarrabi-Zadeh (WAOA2007 and Theory of Computing Systems 45(3), 2009), which is defined as follows: “Points” are given online on the line one by one. An algorithm creates a “cluster,” which is a line segment. The initial length of a cluster is 0, and an algorithm can extend a cluster until it reaches unit length. The task of an algorithm is to cover a new given point either by creating a new cluster and assigning the point to the cluster or by extending an existing cluster created in past times. The goal is to minimize the total number of created clusters. In this paper, we show a lower bound of 13 / 8 = 1.625 on the competitive ratio of any deterministic online algorithm for the one-dimensional unit clustering, improving the previous lower bound 8 / 5 ( = 1.6 ) presented by Epstein and van Stee (WAOA2007 and ACM Transactions on Algorithms 7(1), 2010). Note that Ehmsen and Larsen (SWAT2010 and Theoretical Computer Science, 500, 2013) showed the current best upper bound of 5/3. [ABSTRACT FROM AUTHOR]