Minimum Machine Completion Time Research Articles

On the price of anarchy of two-stage machine scheduling games

We consider a scheduling game, in which both the machines and the jobs are players. Machines are controlled by different selfish agents and attempt to maximize their workloads by choosing a scheduling policy among the given set of policies, while each job is controlled by a selfish agent that attempts to minimize its completion time by selecting a machine. Namely, this game was done in two-stage. In the first stage, every machine simultaneously chooses a policy from some given set of policies, and in the second stage, every job simultaneously chooses a machine. In this work, we use the price of anarchy to measure the efficiency of such equilibria where each machine is allowed to use one of the at most two policies. We provide nearly tight bounds for every combination of two deterministic scheduling policies with respect to two social objectives: minimizing the maximum job completion, and maximizing the minimum machine completion time.

MapReduce machine covering problem on a small number of machines

We study machine covering problem in MapReduce system. Each job consists of two sets of tasks, namely the map tasks and reduce tasks. A job’s reduce tasks can only be processed after all its map tasks are finished. The map tasks are fractional, i.e., they can be arbitrarily split and processed on different machines in parallel. Our goal is to maximize the minimum machine completion time. We consider two variants of the problem, namely the cases involving preemptive reduce tasks and non-preemptive reduce tasks. For preemptive reduce tasks, we present optimal solution algorithms for the problem on two and three machines. For non-preemptive reduce tasks, we provide an approximation algorithm with a tight worse-case ratio of $$\frac{4}{3}$$ for the problem on two machines.

Reduction criteria, upper bounds, and a dynamic programming based heuristic for the max–min $$k_i$$ k i -partitioning problem

This paper addresses the max–min $$k_i$$ -partitioning problem that asks for an assignment of n jobs to m parallel machines so that the minimum machine completion time is maximized and the number of jobs on each machine does not exceed a machine-dependent cardinality limit $$k_i$$ $$(i=1,\ldots ,m)$$ . We propose different preprocessing as well as lifting procedures and derive several upper bound arguments. Furthermore, we introduce suited construction heuristics as well as an effective dynamic programming based improvement procedure. Results of a comprehensive computational study on a large set of randomly generated instances indicate that our algorithm quickly finds (near-)optimal solutions.

Improved approaches to the exact solution of the machine covering problem

For the basic problem of scheduling a set of n independent jobs on a set of m identical parallel machines with the objective of maximizing the minimum machine completion time--also referred to as machine covering--we propose a new exact branch-and-bound algorithm. Its most distinctive components are a different symmetry-breaking solution representation, enhanced lower and upper bounds, and effective novel dominance criteria derived from structural patterns of optimal schedules. Results of a comprehensive computational study conducted on benchmark instances attest to the effectiveness of our approach, particularly for small ratios of n to m.

Parallel machine covering with limited number of preemptions

In this paper, we investigate the i-preemptive scheduling on parallel machines to maximize the minimum machine completion time, i.e., machine covering problem with limited number of preemptions. It is aimed to obtain the worst case ratio of the objective value of the optimal schedule with unlimited preemptions and that of the schedule allowed to be preempted at most i times. For the m identical machines case, we show the worst case ratio is {ie18-1}, and we present a polynomial time algorithm which can guarantee the ratio for any 0 ≤ i ≤ m − 1. For the i-preemptive scheduling on two uniform machines case, we only need to consider the cases of i = 0 and i = 1. For both cases, we present two linear time algorithms and obtain the worst case ratios with respect to s, i.e., the ratio of the speeds of two machines.

Comparing the minimum completion times of two longest-first scheduling-heuristics

For the basic problem of non-preemptively scheduling n independent jobs on m identical parallel machines so that the minimum (or earliest) machine completion time is maximized, we compare the performance relationship between two well-known longest-first heuristics—the LPT- (longest processing time) and the RLPT-heuristic (restricted LPT). We provide insights into the solution structure of these two sequencing heuristics and prove that the minimum completion time of the LPT-schedule is at least as long as the minimum completion time of the RLPT-schedule. Furthermore, we show that the minimum completion time of the RLPT-heuristic always remains within a factor of 1/m of the optimal minimum completion time. The paper finishes with a comprehensive experimental study of the probabilistic behavior of RLPT-schedules compared to LPT-schedules in the two-machine case.

Online and semi-online hierarchical scheduling for load balancing on uniform machines

In this paper, we consider online and semi-online hierarchical scheduling for load balancing on m parallel uniform machines with two hierarchies. There are k machines with speed s and hierarchy 1 which can process all the jobs, while the remaining m − k machines with speed 1 and hierarchy 2 can only process jobs with hierarchy 2. For the model with the objective to maximize the minimum machine completion time, we show that no online algorithm can achieve bounded competitive ratio, i.e., there exists no competitive algorithm. We overcome this barrier by way of fractional assignment, where each job can be arbitrarily split between the machines. We design a best possible algorithm with competitive ratio 2 k s + m − k k s + m − k for any 0 < s < ∞ . For the model with the objective of minimizing makespan, we give a best possible algorithm with competitive ratio ( k s + m − k ) 2 k 2 s 2 + k ( m − k ) s + ( m − k ) 2 for any 0 < s < ∞ . For the semi-online model with the objective to minimize makespan, where the total job size ( s u m ) is known in advance, we present an optimal algorithm (i.e., an algorithm of competitive ratio 1).

POLYNOMIAL APPROXIMATION SCHEMES FOR THE MAX-MIN ALLOCATION PROBLEM UNDER A GRADE OF SERVICE PROVISION

The max-min allocation problem under a grade of service provision is defined in the following model: given a set [Formula: see text] of m parallel machines and a set [Formula: see text] of n jobs, where machines and jobs are all entitled to different levels of grade of service (GoS), each job [Formula: see text] has its processing time pj and it can only be allocated to a machine Mi whose GoS level is no more than the GoS level the job Jj has. The goal is to allocate all jobs to m machines to maximize the minimum machine load among these m machines, where the machine load of Mi is the sum of the processing time of jobs executed on Mi. The best known approximation algorithm [4] to solve this problem produces an allocation in which the minimum machine completion time is at least Ω ( log log log m/ log log m) of the optimal value. In this paper, we respectively present four approximation schemes to solve this problem and its two special versions: (1) a polynomial time approximation scheme (PTAS) with a running time [Formula: see text] for the general version, where ϵ &gt; 0; (2) a PTAS and a fully polynomial time approximation scheme (FPTAS) with the running time O(n) for the version where the number m of machines is fixed; (3) a PTAS with the running time O(n) for the version where the number of GoS levels is bounded by k.

Parallel machine scheduling problems with proportionally deteriorating jobs

This article considers two scheduling problems on parallel identical machines with deteriorating jobs, in which the processing time of job is proportional function of its starting times to be processed. Namely, for job j, p j = b j t, where b j > 0 is its deterioration rate, t ≥ t 0 > 0 is its starting time and t 0 is a given constant. The criteria of optimality are minimising the makespan and maximising the minimum machine completion time. It has been proved that the first problem is NP-hard; we prove that the second problem is also NP-hard. For both problems heuristic algorithms are constructed and their performance is analysed.

Maximizing the minimum completion time on parallel machines

We propose an exact branch-and-bound algorithm for the problem of maximizing the minimum machine completion time on identical parallel machines. The proposed algorithm is based on tight lower and upper bounds as well as an effective symmetry-breaking branching strategy. Computational results performed on a large set of randomly generated instances attest to the efficacy of the proposed algorithm.

Semi-online Machine Covering on Two Uniform Machines with Known Total Size

This paper investigates semi-online scheduling problem with known total size on two uniform machines for maximizing the minimum machine completion time. Lower bounds and optimal algorithms for every s≥1 are given, where s is the speed ratio of two machines. Both the overall competitive ratio and the competitive ratio for Open image in new window are strictly smaller than those of the optimal algorithms of the corresponding semi-online scheduling problem with known optimal value. It indicates that two related semi-online problems are different.

Optimal Preemptive Online Algorithms for Scheduling with Known Largest Size on Two Uniform Machines

In this paper, we consider the semi-online preemptive scheduling problem with known largest job sizes on two uniform machines. Our goal is to maximize the continuous period of time (starting from time zero) when both machines are busy, which is equivalent to maximizing the minimum machine completion time if idle time is not introduced. We design optimal deterministic semi-online algorithms for every machine speed ratio s ∈¸ [1,∞), and show that idle time is required to achieve the optimality during the assignment procedure of the algorithm for any s > (s 2 + 3s + 1 / (s 2 + 2s + 1). The competitive ratio of the algorithms is (s 2 + 3s + 1) / (s 2 + 2s + 1), which matches the randomized lower bound for every s ≥ 1. Hence randomization does not help for the discussed preemptive scheduling problem.

Semi On-Line Scheduling Problem for Maximizing the Minimum Machine Completion Time on Two Uniform Machines

In this paper, we consider a semi on-line version on two uniform machines Mi, i = 1,2, where the processing time of the largest job is known in advance. A speed si(s1 = 1, 1≤ s2 = s) is associated with machine Mi. Our goal is to maximize the Cmin. We give a Cmin 2 algorithm and prove its competitive ratio is at most \(\frac{2s+1}{s+1}\). We also claim the Cmin 2 algorithm is tight and the gap between the competitive ratio of Cmin 2 algorithm and the optimal value is not greater than 0.555. It is obvious that our result coincides with that given by He for s = 1.

Preemptive Semi–online Algorithms for Parallel Machine Scheduling with Known Total Size

This paper investigates preemptive semi-online scheduling problems on m identical parallel machines, where the total size of all jobs is known in advance. The goal is to minimize the maximum machine completion time or maximize the minimum machine completion time. For the first objective, we present an optimal semi–online algorithm with competitive ratio 1. For the second objective, we show that the competitive ratio of any semi–online algorithm is at least \( \frac{{2m - 3}} {{m - 1}} \) for any m > 2 and present optimal semi–online algorithms for m = 2, 3.

Preemptive Machine Covering on Parallel Machines

This paper investigates the preemptive parallel machine scheduling to maximize the minimum machine completion time. We first show the off-line version can be solved in O(mn) time for general m-uniform-machine case. Then we study the on-line version. We show that any randomized on-line algorithm must have a competitive ratio m for m-uniform-machine case and ∑i = 1m1/i for m-identical-machine case. Lastly, we focus on two-uniform-machine case. We present an on-line deterministic algorithm whose competitive ratio matches the lower bound of the on-line problem for every machine speed ratio s≥ 1. We further consider the case that idle time is allowed to be introduced in the procedure of assigning jobs and the objective becomes to maximize the continuous period of time (starting from time zero) when both machines are busy. We present an on-line deterministic algorithm whose competitive ratio matches the lower bound of the problem for every s≥ 1. We show that randomization does not help.

Semi on-line scheduling for maximizing the minimum machine completion time on three uniform machines

The paper investigates a semi on-line scheduling problem wherein the largest processing time of jobs done by three uniform machinesM1,M2,M3 is known in advance. A speedsi(S1=1,s2=r,s3=s, 1≤r≤s) is associated with machineMi. Our goal is to maximizeCmin-the minimum workload of the three machines. We present a min3 algorithm and prove its competitive ratio is max {r+1,(3s+r+1)/(1+r+s)}, with the lower bound being at least max {2,r}. We also claim the competitive ratio of min3 algorithm cannot be improved and is the best possible for 1≤s≤2,r=1.

Semi on-line scheduling for maximizing the minimum machine completion time on three uniform machines

Semi on-line scheduling for maximizing the minimum machine completion time on three uniform machines

Optimal semi-online preemptive algorithms for machine covering on two uniform machines

In this paper, we consider the semi-online preemptive scheduling problem with decreasing job sizes on two uniform machines. Our goal is to maximize the continuous period of time (starting from time zero) when both machines are busy, which is equivalent to maximizing the minimum machine completion time if idle time is not introduced before all the jobs are completed. We design optimal deterministic semi-online algorithms for every machine speed ratio s ∈ [ 1 , ∞ ) , and show that idle time is required during the assignment procedure of algorithms for any s > 6 / 2 . The competitive ratios of the algorithms match the randomized lower bound for every 1 ⩽ s ⩽ 3 . The problem of whether randomization still does not help for the discussed preemptive scheduling problem remains open.

Semi on-line scheduling problem for maximizing the minimum machine completion time on m identical machines

In this paper, a semi cm-line version on m identical machines M1, M2, …, Mm (m≥3) was considered, where the processing time of the largest job is known in advance. Our goal is to maximize the minimum machine load, an NPLS algorithm was presented and its worst-case ratio was proved to be equal to m−1 which is the best possible value. It is concluded that if the total processing time of jobs is also known to be greater than (2m−1)pmax where pmax is the largest job’s processing time, then the worst-case ratio is 2−1/m.

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 l p 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 l p norm, we study the case of identical machines ( s = 1) and present tight bounds as a function of p .