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• We consider a variant of the online buffer management problem. • This variant is called the 2-bounded delay buffer management problem with lookahead. • We design an optimal online algorithm for this variant. The bounded delay buffer management problem, which was proposed by Kesselman et al. (STOC 2001 and SIAM Journal on Computing 33(3), 2004), is an online problem focusing on buffer management of a switch supporting Quality of Service (QoS). The problem definition is as follows: Packets arrive to a buffer over time and each packet is specified by the release time , deadline and value. An algorithm can transmit at most one packet from the buffer at each integer time and can gain its value as the profit if transmitting the packet by its deadline after its release time. The objective of this problem is to maximize the gained profit. We say that an instance of the problem is s -bounded if for any packet, an algorithm has at most s chances to transmit it. For any s ≥ 2 , Hajek (CISS 2001) showed that the competitive ratio of any deterministic algorithm is at least (1 + 5) / 2 ≥ 1.618. Recently, Veselý et al. (SODA 2019) designed an online algorithm matching the lower bound. Böhm et al. (ISAAC 2016 and Theoretical Computer Science, 2019) introduced the lookahead ability to an online algorithm. At a time t , the algorithm obtains information about packets arriving at time t + 1 , and showed that for s = 2 , there is an algorithm which achieves the competitive ratio of (− 1 + 13) / 2 ≤ 1.303. Also, they showed that the competitive ratio of any deterministic algorithm is at least (1 + 17) / 4 ≥ 1.280. In this paper, for the 2-bounded model with lookahead, we design an algorithm with a matching competitive ratio of (1 + 17) / 4. [ABSTRACT FROM AUTHOR]

• We consider a new variant of the online interval scheduling problem. • This variant formulates best-effort requests and responses as an online problem. • We show some upper and lower bounds on the competitive ratios of this variant. The interval scheduling problem is one variant of the scheduling problem. In this paper, we propose a novel variant of the interval scheduling problem, whose definition is as follows: given jobs are specified by their release times , deadlines and profits. An algorithm must start a job at its release time on one of m identical machines, and continue processing until its deadline on the machine to complete the job. All the jobs must be completed and the algorithm can obtain the profit of a completed job as a user's satisfaction. It is possible to process more than one job at a time on one machine. The profit of a job is distributed uniformly between its release time and deadline, that is its interval, and the profit gained from a subinterval of a job decreases in reverse proportion to the number of jobs whose intervals intersect with the subinterval on the same machine. The objective of our variant is to maximize the total profit of completed jobs. This formulation is naturally motivated by best-effort requests and responses to them, which appear in many situations. In best-effort requests and responses, the total amount of available resources for users is always invariant and the resources are equally shared with every user. We study online algorithms for this problem. Specifically, we show that for the case where the profits of jobs are arbitrary, there does not exist an algorithm whose competitive ratio is bounded. Then, we consider the case in which the profit of each job is equal to its length, that is, the time interval between its release time and deadline. For this case, we prove that for m = 2 and m ≥ 3 , the competitive ratios of a greedy algorithm are at most 4/3 and at most 3, respectively. Also, for each m ≥ 2 , we show a lower bound on the competitive ratio of any deterministic algorithm. [ABSTRACT FROM AUTHOR]