Narrow Results

We investigate an online scheduling problem on a bounded batch machine with $f$ incompatible job families, in which the jobs are released over time and the jobs belonging to the same family have the same processing times. The goal is to minimize the maximum completion time. A machine can process at most $b$ jobs simultaneously as a batch, where $b$ is finite. A batch only contains the jobs from the same family. KRT setting means that no job is released when the machine is busy. In this paper, we consider the above model under two environments: (1) KRT setting and (2) general setting. In the KRT setting, we provide the lower bounds $1+\frac{\sqrt{f^2-f+1}-1}{f}$ for $b\ge f$ and $min\left\{\frac{2f+1}{f+2},\frac{2b}{b+1}\right\}$ for $2\le b<f$. In the general setting, we provide the lower bounds $1+\frac{\sqrt{4f^2+1}-1}{2f}$ for $b\ge f+1$ and $\frac{2b}{b+1}$ for $2\le b<f+1$. We further present an online algorithm, which is the best possible when $b\ge f$ for the KRT setting and when $b\ge f+1$ for the general setting.

In this paper, we consider the on-line single machine scheduling of unit time jobs with rejection. All jobs arrive on-line over a list (one by one). For each arriving job, the on-line algorithm must decide immediately to accept or reject it. The objective is to minimize the maximum quadratic completion time of accepted jobs plus the total rejection cost of rejected jobs. For this problem, we show that 1.7299 is a lower bound on the competitive ratio and present a simple greedy algorithm with the competitive ratio 2. Furthermore, we also provide a modified greedy algorithm with a better competitive ratio $1+\frac{\sqrt{3}}{2}\approx 1.86602$.

In this paper, some results concerning the k-truck problem are produced. Firstly, the algorithms and their complexity concerning the off-line k-truck problem are discussed. Following that, a lower bound of competitive ratio (1+θ)·k/(θ·k+2) for the on-line k-truck problem is given, where θ is the ratio of cost of the loaded truck and the empty truck on the same distance, and a relevant lower bound for the on-line k-taxi problem followed naturally. Thirdly, based on the Position Maintaining Strategy (PMS), some new results which are slightly better than those of [11] for general cases are obtained. For example, a c-competitive or (c/θ+1/θ+1)-competitive algorithm for the on-line k-truck problem depending on the value of θ, where c is the competitive ratio of some algorithm to a relevant k-server problem, is developed. The Partial-Greedy Algorithm (PG) is used as well to solve this problem on a line with n nodes and is proved to be a (1+(n-k)/θ)-competitive algorithm for this case. Finally, the concepts of the on-line k-truck problem are extended to obtain a new variant: Deeper On-line k-Truck Problem (DTP). We claim that results of PMS for the STP (Standard Truck Problem) hold for the DTP.

In this paper we study an online minimum makespan scheduling problem with a reordering buffer. We obtain the following results: (i) for m > 51 identical machines, we give a 1.5-competitive online algorithm with a buffer of size ⌈1.5m⌉; (ii) for three identical machines, we give an optimal online algorithm with a buffer size six, better than the previous nine; (iii) for m uniform machines, using a buffer of size m, we improve the competitive ratio from 2 + ε to 2 − 1/m+ ε, where ε > 0 is sufficiently small and m is a constant.

We consider the online scheduling problem in a CPU-GPU cluster. In this problem there are two sets of processors, the CPU processors and the GPU processors. Each job has two distinct processing times, one for the CPU processor and the other for the GPU processor. Once a job is released, a decision should be made immediately about which processor it should be assigned to. The goal is to minimize the makespan, i.e., the largest completion time among all the processors. Such a problem could be seen as an intermediate model between the scheduling problem on identical machines and unrelated machines. We provide a 3.85-competitive online algorithm for this problem and show that no online algorithm exists with competitive ratio strictly less than 2. We also consider two special cases of this problem, the balanced case where the number of CPU processors equals to that of GPU processors, and the one-sided case where there is only one CPU or GPU processor. For the balanced case, we first provide a simple 3-competitive algorithm, and then a better algorithm with competitive ratio of 2.732 is derived. For the one-sided case, a 3-competitive algorithm is given.

This work investigates an online two stage k-search problem where an online player makes selections in two stages. In the first stage a number of more than k quoted prices are selected as candidates, and then exactly k highest quoted prices are chosen from the candidates in the second stage. The objective is to maximize the total profit of the k final accepted prices. We mainly propose a deterministic online algorithm and prove that it is optimal in competitiveness. A further discussion is given considering various relationships between the value of k and the number of candidates.

In an online problem, the input is revealed one piece at a time. In every time step, the online algorithm has to produce a part of the output, based on the partial knowledge of the input. Such decisions are irrevocable, and thus online algorithms usually lead to nonoptimal solutions. The impact of the partial knowledge depends strongly on the problem. If the algorithm is allowed to read binary information about the future, the amount of bits read that allow the algorithm to solve the problem optimally is the so-called advice complexity. The quality of an online algorithm is measured by its competitive ratio, which compares its performance to that of an optimal offline algorithm.

In this paper we study online bipartite matchings focusing on the particular case of bipartite matchings in regular graphs. We give tight upper and lower bounds on the competitive ratio of the online deterministic bipartite matching problem. The competitive ratio turns out to be asymptotically equal to the known randomized competitive ratio. Afterwards, we present an upper and lower bound for the advice complexity of the online deterministic bipartite matching problem.

We consider the on-line problem of scheduling n independent jobs on m identical machines under the machine eligibility constraints, where each job has its own specified subset of machines which are eligible for processing it. We investigate a greedy algorithm LS and prove its posterior competitiveness ratio is , where λ is the number of machines eligible for processing the job with the latest completion time.

In this paper, we investigate a semi-on-line version for a special case of three machines M₁, M₂, M₃ where the processing time of the largest job is known in advance. A speed s_i(s₁ = s₂ = 1, 1 ≤ s₃ = s) is associated with machine M_i. Our goal is to maximize the C_min — the minimum workload of three machines. We give a C_min3 algorithm and prove its competitive ratio is and the algorithm is the best possible for 1 ≤ s ≤ 2. We also claim the competitive ratio of algorithm C_min3 is tight.

In this paper, we consider a semi-online preemptive scheduling problem on two uniform machines, where we assume that all jobs have sizes between p and rp for some p > 0 and r ≥ 1. The goal is to maximize the continuous period of time (starting from time zero) when both machines are busy. We present an optimal semi-online algorithm for any combination of the job size ratio r and machine speed ratio s.

This paper investigates two different semi-online versions of the machine covering, which is the problem of assigning a set of jobs to a system of m(m ≥ 3) identical parallel machines so as to maximize the earliest machine completion time. In the first case, we assume that the largest processing times is known in advance. In the second case, we assume that the total processing times of all jobs is known in advance. For each version we propose a semi-online algorithm and investigate its competitive ratio. The competitive ratio of each algorithm is , which is shown to be the best possible competitive ratio for each semi-online problem.

In this paper, we consider a semi-on-line scheduling problem of two identical machines with common maintenance time interval and nonresumable availability. We prove a lower bound of 2.79129 on the competitive ratio and give an on-line algorithm with competitive ratio 2.79633 for this problem.

We consider online scheduling of unit length jobs on m identical parallel-batch machines. Jobs arrive over time. The objective is to minimize maximum flow-time, with the flow-time of a job being the difference of its completion time and its release time. A parallel-batch machine can handle up to b jobs simultaneously as a batch. Here, the batch capacity is bounded, that is b < ∞. In this paper, we provide a best possible online algorithm for the problem with a competitive ratio of .

In this paper, we consider the online-list scheduling on a single bounded parallel-batch machine to minimize makespan. In the problem, the jobs arrive online over list. The first unassigned job in the list should be assigned to a batch before the next job is released. Each batch can accommodate up to b jobs. For b = 2, we establish a lower bound 1 + γ of competitive ratio and provide an online algorithm with a competitive ratio of , where γ is the positive root of γ(γ + 1)² = 1. For b = 3, we establish a lower bound 1 + α of competitive ratio and provide an online algorithm with a competitive ratio of 2, where α is the positive root of the equation (1 + α)(1 + α²) = 2.

In this paper, we consider the online scheduling on m identical machines in which jobs arrive over time. The goal is to determine a nonpreemptive schedule that minimizes the weighted makespan, i.e., the maximum weighted completion time of jobs. When m = 1, we first present a lower bound 2, and then provide an online algorithm with a competitive ratio of 3. For the case in which m ≥ 1, and all jobs have a common processing time p > 0, we obtain a best possible online algorithm with a competitive ratio of .

This paper studies the online hierarchical scheduling problem on two uniform machines with bounded job sizes, where the first machine M₁ receives both low and high hierarchy jobs, while the second machine M₂ only receives high hierarchy jobs. The machines have a speed ratio of s(s ≥ 1), and M₂ runs faster. Jobs are revealed one by one, and before the current job is scheduled, we have no information about next jobs except that the size of any job is in the interval [1, t]. The objective is to minimize the makespan. We present optimal algorithms for all (s, t) pairs.

This paper addresses online scheduling of malleable parallel jobs to minimize the maximum completion time, i.e., makespan. It is assumed that the execution time of a job J_j with processing time p_j is p_j/k + (k-1)c if the job is assigned to k machines, where c > 0 is a constant setup time. We consider online algorithms for the scheduling problem on two identical machines. Namely, the job J_j can be processed on one machine with execution time p_j or alternatively two machines in parallel with execution time p_j/2+c. For the asymptotical competitive ratio, we provide an improved online algorithm with makespan no more than (3/2)C* +c/2, where C* is the optimal makespan. For the strict competitive ratio, we propose an online algorithm with competitive ratio of 1.54, which is close to the lower bound of 1.5.

In this paper, we consider the online single machine scheduling problem to minimize the maximum starting time of the jobs. For the non-preemptive model, we show that there is no determined or randomized online algorithm with a bounded competitive ratio. For the preemption-resume model, we show that the well-known SRPT rule yields an optimal schedule. For the preemption-restart model, we show that any determined online algorithm has a competitive ratio of at least 2 and present an online algorithm with the best-possible competitive ratio of 2.

In this paper, we study an online scheduling on two parallel machines in MapReduce-like system where each job contains two kinds of tasks: map tasks and reduce tasks. A job’s reduce tasks can only be processed after all its map tasks are finished. We assume that the map tasks are fractional and the reduce tasks are preemptive. Our objective is to minimize makespan. We show that the lower bound for this MapReduce scheduling problem is $\sqrt{2}$. We then present an online algorithm with competitive ratio of $\sqrt{2}$ and thus it is optimal.

In this paper, we consider the online scheduling of incompatible family jobs with equal length on an unbounded parallel-batch machine with job delivery. The jobs arrive online over time and belong to $f$ incompatible job families, where $f$ is known in advance. The jobs are first processed in batches on an unbounded parallel-batch machine and then the completed jobs are delivered in batches by a vehicle with infinite capacity to their customers. The jobs from distinct families cannot be processed and delivered in the same batch. The objective is to minimize the maximum delivery completion time of the jobs. For this problem, we present an online algorithm with the best competitive ratio of $1+\frac{\sqrt{4f^2+1}-1}{2f}$.