Multi-phase Algorithms for Throughput Maximization for Real-Time Scheduling

We consider the problem of off-line throughput maximization for job scheduling on one or more machines, where each job has a release time, a deadline and a profit. Most of the versions of the problem discussed here were already treated by Bar-Noy et al. (Proc. 31st ACM STOC, 1999, pp. 622–631; http://www.eng.tau.ac.il/amotz/). Our main contribution is to provide algorithms that do not use linear programming, are simple and much faster than the corresponding ones proposed in Bar-Noy et al. (ibid., 1999), while either having the same quality of approximation or improving it. More precisely, compared to the results of in Bar-Noy et al. (ibid., 1999), our pseudo-polynomial algorithm for multiple unrelated machines and all of our strongly-polynomial algorithms have better performance ratios, all of our algorithms run much faster, are combinatorial in nature and avoid linear programming. Finally, we show that algorithms with better performance ratios than 2 are possible if the stretch factors of the jobs are bounded; a straightforward consequence of this result is an improvement of the ratio of an optimal solution of the integer programming formulation of the JISP2 problem (see Spieksma, Journal of Scheduling, vol. 2, pp. 215–227, 1999) to its linear programming relaxation.     

Improved approximation algorithms for non-preemptive multiprocessor scheduling with testing

Multiprocessor scheduling, also called scheduling on parallel identical machines to minimize the makespan, is a classic optimization problem which has been extensively studied. Scheduling with testing is an online variant, where the processing time of a job is revealed by an extra test operation, otherwise the job has to be executed for a given upper bound on the processing time. Albers and Eckl recently studied the multiprocessor scheduling with testing; among others, for the non-preemptive setting they presented an approximation algorithm with competitive ratio approaching 3.1016 when the number of machines tends to infinity and an improved approximation algorithm with competitive ratio approaching 3 when all test operations take one unit of time each. We propose to first sort the jobs into non-increasing order of the minimum value between the upper bound and the testing time, then partition the jobs into three groups and process them group by group according to the sorted job order. We show that our algorithm achieves better competitive ratios, which approach 2.9513 when the number of machines tends to infinity in the general case; when all test operations each takes one time unit, our algorithm achieves even better competitive ratios approaching 2.8081.

     

A simple linear time approximation algorithm for multi-processor job scheduling on four processors

Multiprocessor job scheduling problem has become increasingly interesting, for both theoretical study and practical applications. Theoretical study of the problem has made significant progress recently, which, however, seems not to imply practical algorithms for the problem, yet. Practical algorithms have been developed only for systems with three processors and the techniques seem difficult to extend to systems with more than three processors. This paper offers new observations and introduces new techniques for the multiprocessor job scheduling problem on systems with four processors. A very simple and practical linear time approximation algorithm of ratio bounded by 1.5 is developed for the multi-processor job scheduling problem P ₄|fix|C _max, which significantly improves previous results. Our techniques are also useful for multiprocessor job scheduling problems on systems with more than four processors.     

Online scheduling on parallel machines with two GoS levels

This paper investigates the online scheduling problem on parallel and identical machines with a new feature that service requests from various customers are entitled to many different grade of service (GoS) levels. Hence each job and machine are labeled with the GoS levels, and each job can be processed by a particular machine only when the GoS level of the job is not less than that of the machine. The goal is to minimize the makespan. In this paper, we consider the problem with two GoS levels. It assumes that the GoS levels of the first k machines and the last m−k machines are 1 and 2, respectively. And every job has a GoS level of 1 alternatively or 2. We first prove the lower bound of the problem under consideration is at least 2. Then we discuss the performance of algorithm AW presented in Azar et al. (J. Algorithms 18:221–237, 1995) for the problem and show it has a tight bound of 4−1/m. Finally, we present an approximation algorithm with competitive ratio . Research supported by Natural Science Foundation of Zhejiang Province (Y605316) and its preliminary version appeared in Proceedings of AAIM2006, LNCS, 4041, 11-21.     

Semi-online scheduling with “end of sequence” information

We study a variant of classical scheduling, which is called scheduling with “end of sequence” information. It is known in advance that the last job has the longest processing time. Moreover, the last job is marked, and thus it is known for every new job whether it is the final job of the sequence. We explore this model on two uniformly related machines, that is, two machines with possibly different speeds. Two objectives are considered, maximizing the minimum completion time and minimizing the maximum completion time (makespan). Let s be the speed ratio between the two machines, we consider the competitive ratios which are possible to achieve for the two problems as functions of s. We present algorithms for different values of s and lower bounds on the competitive ratio. The proposed algorithms are best possible for a wide range of values of s. For the overall competitive ratio, we show tight bounds of ϕ + 1 ≈ 2.618 for the first problem, and upper and lower bounds of 1.5 and 1.46557 for the second problem. The authors would like to dedicate this paper to the memory of our colleague and friend Yong He who passed away in August 2005 after struggling with illness. D. Ye: Research was supported in part by NSFC (10601048).     

2-Distance vertex-distinguishing index of subcubic graphs

This paper addresses the performance of scheduling algorithms for a two-stage no-wait hybrid flowshop environment with inter-stage flexibility, where there exist several parallel machines at each stage. Each job, composed of two operations, must be processed from start to completion without any interruption either on or between the two stages. For each job, the total processing time of its two operations is fixed, and the stage-1 operation is divided into two sub-parts: an obligatory part and an optional part (which is to be determined by a solution), with a constraint that no optional part of a job can be processed in parallel with an idleness of any stage-2 machine. The objective is to minimize the makespan. We prove that even for the special case with only one machine at each stage, this problem is strongly NP-hard. For the case with one machine at stage 1 and m machines at stage 2, we propose two polynomial time approximation algorithms with worst case ratio of \(3-\frac{2}{m+1}\) and \(2-\frac{1}{m+1}\), respectively. For the case with m machines at stage 1 and one machine at stage 2, we propose a polynomial time approximation algorithm with worst case ratio of 2. We also prove that all the worst case ratios are tight.     

Approximations to Optimal k‐Unit Cycles for Single‐Gripper and Dual‐Gripper Robotic Cells

We consider the problem of scheduling operations in bufferless robotic cells that produce identical parts using either single‐gripper or dual‐gripper robots. The objective is to find a cyclic sequence of robot moves that minimizes the long‐run average time to produce a part or, equivalently, maximizes the throughput. Obtaining an efficient algorithm for an optimum k‐unit cyclic solution (k ≥ 1) has been a longstanding open problem. For both single‐gripper and dual‐gripper cells, the approximation algorithms in this paper provide the best‐known performance guarantees (obtainable in polynomial time) for an optimal cyclic solution. We provide two algorithms that have a running time linear in the number of machines: for single‐gripper cells (respectively, dual‐gripper cells), the performance guarantee is 9/7 (respectively, 3/2). The domain considered is free‐pickup cells with constant intermachine travel time. Our structural analysis is an important step toward resolving the complexity status of finding an optimal cyclic solution in either a single‐gripper or a dual‐gripper cell. We also identify optimal cyclic solutions for a variety of special cases. Our analysis provides production managers valuable insights into the schedules that maximize productivity for both single‐gripper and dual‐gripper cells for any combination of processing requirements and physical parameters.     

Scheduling with release times and rejection on two parallel machines

In this paper we study scheduling with release times and job rejection on two parallel machines. In our scheduling model each job is either accepted and then processed by one of the two machines at or after its release time, or it is rejected and then a rejection penalty is paid. The objective is to minimize the makespan of the accepted job plus the total penalty of all rejected jobs. The scheduling problem is NP-hard in the ordinary sense. In this paper, we develop a \(1.5+\epsilon \)-approximation algorithm for the problem, where \(\epsilon \) is any given small positive constant.     

Semi-online scheduling on 2 machines under a grade of service provision with bounded processing times

We study the problem of semi-online scheduling on 2 machines under a grade of service (GoS). GoS means that some jobs have to be processed by some machines to be guaranteed a high quality. The problem is online in the sense that jobs are presented one by one, and each job shall be assigned to a time slot on its arrival. Assume that the processing time p _i of every job J _i is bounded by an interval [a,α a], where a>0 and α>1 are two constant numbers. By knowing the bound of jobs’ processing times, we denote it by semi-online problem. We deal with two semi-online problems.     

Online scheduling with a buffer on related machines

Online scheduling with a buffer is a semi-online problem which is strongly related to the basic online scheduling problem. Jobs arrive one by one and are to be assigned to parallel machines. A buffer of a fixed capacity K is available for storing at most K input jobs. An arriving job must be either assigned to a machine immediately upon arrival, or it can be stored in the buffer for unlimited time. A stored job which is removed from the buffer (possibly, in order to allocate a space in the buffer for a new job) must be assigned immediately as well. We study the case of two uniformly related machines of speed ratio s≥1, with the goal of makespan minimization.     

On-Line Scheduling a Batch Processing System to Minimize Total Weighted Job Completion Time

Scheduling a batch processing system has been extensively studied in the last decade. A batch processing system is modelled as a machine that can process up to b jobs simultaneously as a batch. The scheduling problem involves assigning all n jobs to batches and determining the batch sequence in such a way that certain objective function of job completion times C _j is minimized. In this paper, we address the scheduling problem under the on-line setting in the sense that we construct our schedule irrevocably as time proceeds and do not know of the existence of any job that may arrive later. Our objective is to minimize the total weighted completion time w _j C _j. We provide a linear time on-line algorithm for the unrestrictive model (i.e., b n) and show that the algorithm is 10/3-competitive. For the restrictive model (i.e., b < n), we first consider the (off-line) problem of finding a maximum independent vertex set in an interval graph with cost constraint (MISCP), which is NP-hard. We give a dual fully polynomial time approximation scheme for MISCP, which leads us to a (4 + )-competitive on-line algorithm for any > 0 for the original on-line scheduling problem. These two on-line algorithms are the first deterministic algorithms of constant performance guarantees.     

Online scheduling on uniform machines with two hierarchies

In this paper we study online scheduling problem on m parallel uniform machines with two hierarchies. The objective is to minimize the maximum completion time (makespan). Machines are provided with different capability. The machines with speed s can schedule all jobs, while the other machines with speed 1 can only process partial jobs. Online algorithms for any 0<s<∞ are provided in the paper. For the case of k=1 and m=2, and the case of some values of s, k=1 and m=3, the algorithms are the best possible, where k is the number of machines with hierarchy 1, and m is the number of machines. Lower bounds for some special cases are also presented.     

Dynamic scheduling of manufacturing job shops using extreme value theory

Most job shop scheduling approaches reported in the literature assume that the scheduling problem is static (i.e. job arrivals and the breakdowns of machines are neglected) and in addition, these scheduling approaches may not address multiple criteria scheduling or accommodate alternate resources to process a job operation. In this paper, a scheduling method based on extreme value theory (SEVAT) is developed and addresses all the shortcomings mentioned above. The SEVAT approach creates a statistical profile of schedules through random sampling, and predicts the quality or 'potential' of a feasible schedule. A dynamic scheduling problem was designed to reflect a real job shop scheduling environment closely. Two performance measures, viz. mean job tardiness and mean job cost, were used to demonstrate multiple criteria scheduling. Three factors were identified, and varied between two levels each, thereby spanning a varied job shop environment. The results of this extensive simulation study show that the SEVAT scheduling approach produces a better performance compared to several common dispatching rules.     

Almost optimal solutions for bin coloring problems

In this paper we study two interesting bin coloring problems: Minimum Bin Coloring Problem (MinBC) and Online Maximum Bin Coloring Problem (OMaxBC), motivated from several applications in networking. For the MinBC problem, we present two near linear time approximation algorithms to achieve almost optimal solutions, i.e., no more than OPT+2 and OPT+1 respectively, where OPT is the optimal solution. For the OMaxBC problem, we first introduce a deterministic 2-competitive greedy algorithm, and then give lower bounds for any deterministic and randomized (against adaptive offline adversary) online algorithms. The lower bounds show that our deterministic algorithm achieves the best possible competitive ratio. The research of this paper was partially supported by an NSF CAREER award CCF-0546509.     

Single-machine scheduling problems with general truncated sum-of-actual-processing-time-based learning effect

We study single machine scheduling problems with general truncated sum-of-actual-processing-time-based learning effect. In the general truncated learning model, the actual processing time of a job is affected by the sum of actual processing times of previous jobs and by a job-dependent truncation parameter. We show that the single machine problems to minimize makespan and to minimize the sum of weighted completion times are both at least ordinary NP-hard and the single machine problem to minimize maximum lateness is strongly NP-hard. We then show polynomial solvable cases and approximation algorithms for these problems. Computational experiments are also conducted to show the effectiveness of our approximation algorithms.

     

On minimum <Emphasis Type="Italic">m</Emphasis>-connected <Emphasis Type="Italic">k</Emphasis>-dominating set problem in unit disc graphs

Minimum m-connected k-dominating set problem is as follows: Given a graph G=(V,E) and two natural numbers m and k, find a subset S⊆V of minimal size such that every vertex in V∖S is adjacent to at least k vertices in S and the induced graph of S is m-connected. In this paper we study this problem with unit disc graphs and small m, which is motivated by the design of fault-tolerant virtual backbone for wireless sensor networks. We propose two approximation algorithms with constant performance ratios for m≤2. We also discuss how to design approximation algorithms for the problem with arbitrarily large m. This work was supported in part by the Research Grants Council of Hong Kong under Grant No. CityU 1165/04E, the National Natural Science Foundation of China under Grant No. 70221001, 10531070 and 10771209.     

An FPTAS for uniform machine scheduling to minimize makespan with linear deterioration

This paper consider m uniform (parallel) machine scheduling with linear deterioration to minimize the makespan. In an uniform machine environment, all machines have different processing speeds. Linear deterioration means that job’s actual processing time is a linear increasing function on its execution starting time. We propose a fully polynomial-time approximation scheme (FPTAS) to show the problem is NP-hard in the ordinary sense.     

Efficient point coverage in wireless sensor networks

We study minimum-cost sensor placement on a bounded 3D sensing field, R, which comprises a number of discrete points that may or may not be grid points. Suppose we have ℓ types of sensors available with different sensing ranges and different costs. We want to find, given an integer σ ≥ 1, a selection of sensors and a subset of points to place these sensors such that every point in R is covered by at least σ sensors and the total cost of the sensors is minimum. This problem is known to be NP-hard. Let k_i denote the maximum number of points that can be covered by a sensor of the ith type. We present in this paper a polynomial-time approximation algorithm for this problem with a proven approximation ratio . In applications where the distance of any two points has a fixed positive lower bound, each k_i is a constant, and so we have a polynomial-time approximation algorithms with a constant guarantee. While γ may be large, we note that it is only a worst-case upper bound. In practice the actual approximation ratio is small, even on randomly generated points that do not have a fixed positive minimum distance between them. We provide a number of numerical results for comparing approximation solutions and optimal solutions, and show that the actual approximation ratios in these examples are all less than 3, even though γ is substantially larger. This research was supported in part by NSF under grant CCF-04080261 and by NSF of China under grant 60273062.     

Efficient Algorithms for a Scheduling Problem and its Applications to Illicit Drug Market Crackdowns

We give polynomial time algorithms for a job scheduling problem. By duality we transform a special case of the drug market crackdown schedulingproblem to the above job scheduling problem and thus derive polynomial timealgorithms to the second problem. Finally, using the algorithm for the specialcase, we develop a quasipolynomial time approximation algorithm for thegeneral case of the drug market crackdown scheduling problem with monomialcost functions.     

Approximation algorithm for the parallel-machine scheduling problem with release dates and submodular rejection penalties

In this paper, we consider the parallel-machine scheduling problem with release dates and submodular rejection penalties. In this problem, we are given m identical parallel machines and n jobs. Each job has a processing time and a release date. A job is either rejected, in which case a rejection penalty has to be paid, or accepted and processed on one of the m identical parallel machines. The objective is to minimize the sum of the makespan of the accepted jobs and the rejection penalty of the rejected jobs which is determined by a submodular function. Our main work is to design a 2-approximation algorithm based on the primal-dual framework.