Zhiyi Tan

Online parallel machines scheduling with two hierarchies

This paper investigates an online hierarchical scheduling problem on m parallel identical machines. Each job, as well as each machine, has a hierarchy associated with it. A job can be scheduled on a machine only when its hierarchy is no higher than that of the machine. The objective is to minimize the makespan. In addition, we assume that there are only two hierarchies, and k machines have a higher hierarchy which can schedule all jobs. We present an online algorithm with a competitive ratio of 1+m^2-mm^2-km+k^2<73 for any k and m. The performance for some pairs of k and m is further improved by another algorithm. Lower bounds for various pairs of k and m are also presented.

Online hierarchical scheduling: An approach using mathematical programming

This paper considers an online hierarchical scheduling problem on parallel identical machines. We are given a set of m machines and a sequence of jobs. Each machine has a different hierarchy, and each job also has a hierarchy associated with it. A job can be assigned to a machine only if its hierarchy is no less than that of the machine. The objective is to minimize the makespan, i.e., the maximum load of all machines. Two models are studied in the paper. For the fractional model, we present an improved algorithm and lower bounds. Both the algorithm and the lower bound are based on solutions of mathematical programming. For any given m, our algorithm is optimal by numerical calculation. For the integral model, we present both a general algorithm for any m, and an improved algorithm with better competitive ratios of 2.333 and 2.610 for m=4 and 5, respectively.

Optimal semi-online algorithms for machine covering

This paper investigates the semi-online machine covering problems on m>=3 parallel identical machines. Three different semi-online versions are studied and optimal algorithms are proposed. We prove that if the total processing time of all jobs or the largest processing time of all jobs is known in advance, the competitive ratios of the optimal algorithms are both m-1. If the total processing time and the largest processing time of all jobs are both known in advance, the competitive ratios of the optimal algorithms are 32 when m=3, and m-2 when m>=4.

Optimal on-line algorithms for the uniform machine scheduling problem with ordinal data

In this paper, we consider an ordinal on-line scheduling problem. A sequence of n independent jobs has to be assigned non-preemptively to two uniformly related machines. We study two objectives which are maximizing the minimum machine completion time, and minimizing the lp, norm of the completion times. It is assumed that the values of the processing times of jobs are unknown at the time of assignment. However it is known in advance that the processing times of arriving jobs are sorted in a non-increasing order. We are asked to construct an assignment of all jobs to the machines at time zero, by utilizing only ordinal data rather than actual magnitudes of jobs. For the problem of maximizing the minimum completion time we first present a comprehensive lower bound on the competitive ratio, which is a piecewise function of machine speed ratio s. Then, we propose an algorithm which is optimal for any s ≥ 1. For minimizing the lp norm, we study the case of identical machines (s = 1) and present tight bounds as a function of p.

Two semi-online scheduling problems on two uniform machines

This paper considers two semi-online scheduling problems, one with known optimal value and the other with known total sum, on two uniform machines with a machine speed ratio of s≥1. For the first problem, we provide an optimal algorithm for s∈[1+√3/2, 1+√21/4], and improved algorithms or/and lower bounds for s∈[1+√21/4, √3], over which the optimal algorithm is unknown. As a result, the largest gap between the competitive ratio and the lower bound decreases to 0.02192. For the second problem, we also present algorithms and lower bounds for s≥1. The largest gap between the competitive ratio and the lower bound is 0.01762, and the length of the interval over which the optimal algorithm is unknown is 0.47382. Our algorithms and lower bounds for these two problems provide insights into their differences, which are unusual from the viewpoint of the known results on these two semi-online scheduling problems in the literature.

Inefficiency of Nash Equilibrium for scheduling games with constrained jobs: A parametric analysis

In this paper, we revisit the inefficiency of Nash Equilibrium of scheduling games by considering the Price of Anarchy (PoA) as a function of r, which is the ratio between the maximum and minimum size of jobs. For the social costs of minimizing makespan and maximizing the minimum machine load of all machines, we obtain the PoA for two and three machines, and the bound is tight for any r>=1. Lower bounds on the PoA for general number of machines are also presented.

Inefficiency of equilibria for scheduling game with machine activation costs

In this paper, we study the scheduling game with machine activation costs. A set of jobs is to be processed on parallel identical machines. The number of machines available is unlimited, and an activation cost is needed whenever a machine is activated in order to process jobs. Each job chooses a machine on which it wants to be processed. The cost of a job is the sum of the load of the machine it chooses and its shared activated cost. The social cost is the total cost of all jobs. Representing the Price of Anarchy (PoA) and Price of Stability (PoS) as functions of the number of jobs, we get the tight bounds of PoA and PoS. Representing PoA and PoS as functions of the smallest processing time of jobs, asymptotically tight bound of PoA and improved lower and upper bounds of PoS are also given.

Coordination mechanisms for scheduling games with proportional deterioration

Abstract We study parallel-machine scheduling games with deteriorating jobs. The processing time of a job increases proportionally with its starting time by a positive deterioration rate. Each job acts selfishly aiming to minimize its completion time while choosing a machine on which it will be processed. Machines are equipped with coordination mechanisms to diminish chaos caused by jobs’ competition. We consider three coordination mechanisms in this paper, namely Smallest Deterioration Rate first, Largest Deterioration Rate first and MAKESPAN policy. Under these mechanisms, we precisely quantify the inefficiency of their Nash Equilibriums by investigating the Price of Anarchy (PoA) and the Price of Stability (PoS), concerning minimization of social costs including the makespan and the total machine load. By using some new methods, we obtain parametrical bounds on the PoA and PoS, and demonstrate that most of these bounds are tight.

The PoA of Scheduling Game with Machine Activation Costs

In this paper, we study the scheduling game with machine activation costs. A set of jobs is to be processed on identical parallel machines. The number of machines available is unlimited, and an activation cost is needed whenever a machine is activated in order to process jobs. Each job chooses a machine on which it wants to be processed. The cost of a job is the sum of the load of the machine it chooses and its shared activated cost. The social cost is the total cost of all jobs. Representing PoA as a function of the number of jobs, we get the tight bound of PoA. Representing PoA as a function of the smallest processing time of jobs, improved lower and upper bound are also given.

Semi-online machine covering for two uniform machines

The machine covering problem deals with partitioning a sequence of jobs among a set of machines, so as to maximize the completion time of the least loaded machine. We study a semi-online variant, where jobs arrive one by one, sorted by non-increasing size. The jobs are to be processed by two uniformly related machines, with a speed ratio of q>=1. Each job has to be processed continuously, in a time slot assigned to it on one of the machines. This assignment needs to be performed upon the arrival of the job. The length of the time slot, which is required for a specific job to run on a given machine, is equal to the size of the job divided by the speed of the machine. We give a complete competitive analysis of this problem by providing an algorithm of the best possible competitive ratio for every q>=1. We first give a tight analysis of the performance of a natural greedy algorithm LPT for the problem. To achieve the best possible performance for the semi-online problem, we use a combination of LPT, together with two alternative algorithms which we design. The new algorithms attain the best possible competitive ratios in the two intervals q@?(1,1.5) and q@?(2.4856,1+3), respectively, whereas the greedy algorithm has the best possible competitive ratio for any other q>=1.