极小化总加权完工时间的Dial-a-Ride问题的在线随机算法

讨论一般度量空间上带单服务器的极小化总加权完工时间在线Dial-a-Ride问题.通过应用贪婪区间的技巧,提出了一个一般在线随机算法.根据这个算法,对于容量为1或者任意容量的一般度量空间上的在线Dial-a-Ride问题能得到一个竞争比为(2+√2)/ln(1+√2)的在线随机算法,这个算法不仅具有当前最好的竞争比,而且也改进了Krumke等人的结果.     

On-line scheduling of unit time jobs with rejection: minimizing the total completion time

We consider on-line scheduling of unit time jobs on a single machine with job-dependent penalties. The jobs arrive on-line (one by one) and can be either accepted and scheduled, or be rejected at the cost of a penalty. The objective is to minimize the total completion time of the accepted jobs plus the sum of the penalties of the rejected jobs.We give an on-line algorithm for this problem with competitive ratio . Moreover, we prove that there does not exist an on-line algorithm with competitive ratio better than 1.63784.     

具有特殊工件的平行机在线排序问题

本文研究一类具有特殊工件的平行机在线排序问题,目标是最小化最大完工时间.此模型有两种工件:正常工件和特殊工件.正常工件能够在m台平行机的任何一台机器上加工,而特殊工件仅能够在它唯一被指定的机器上加工.文中所有特殊工件的指定机器为M1.我们提供了竞争比为(2m2-2m 1)/(m2-m 1)的在线近似算法.当m=2时,算法是最好可能的.当m=3时,算法的竞争比为13/7≈1.857,并且提供了竞争比的下界(1 (平方根33))14≈1.686.     

On-line steiner trees in the Euclidean plane

Suppose we are given a sequence ofn points in the Euclidean plane, and our objective is to construct, on-line, a connected graph that connects all of them, trying to minimize the total sum of lengths of its edges. The points appear one at a time, and at each step the on-line algorithm must construct a connected graph that contains all current points by connecting the new point to the previously constructed graph. This can be done by joining the new point (not necessarily by a straight line) to any point of the previous graph (not necessarily one of the given points). The performance of our algorithm is measured by its competitive ratio: the supremum, over all sequences of points, of the ratio between the total length of the graph constructed by our algorithm and the total length of the best Steiner tree that connects all the points. There are known on-line algorithms whose competitive ratio isO(logn) even for all metric spaces, but the only lower bound known is of [IW] for some contrived discrete metric space. Moreover, for the plane, on-line algorithms could have been more powerful and achieve a better competitive ratio, and no nontrivial lower bounds for the best possible competitive ratio were known. Here we prove an almost tight lower bound of Ω(logn/log logn) for the competitive ratio of any on-line algorithm. The lower bound holds for deterministic algorithms as well as for randomized ones, and obviously holds in any Euclidean space of dimension greater than 2 as well. Noga Alon was supported in part by a USA-Israeli BSF grant.     

On-line supply chain scheduling problems with preemption

We consider supply chain scheduling problems where customers release jobs to a manufacturer that has to process the jobs and deliver them to the customers. The jobs are released on-line, that is, at any time there is no information on the number, release and processing times of future jobs; the processing time of a job becomes known when the job is released. Preemption is allowed. To reduce the total costs, processed jobs are grouped into batches, which are delivered to customers as single shipments; we assume that the cost of delivering a batch does not depend on the number of jobs in the batch. The objective is to minimize the total cost, which is the sum of the total flow time and the total delivery cost. For the single-customer problem, we present an on-line two-competitive algorithm, and show that no other on-line algorithm can have a better competitive ratio. We also consider an extension of the algorithm for the case of m customers, and show that its competitive ratio is not greater than 2m if the delivery costs to different customers are equal.     

Resource augmented semi-online bounded space bin packing

We study on-line bounded space bin-packing in the resource augmentation model of competitive analysis. In this model, the on-line bounded space packing algorithm has to pack a list L of items with sizes in (0, 1], into a minimum number of bins of size b, b≥1. A bounded space algorithm has the property that it only has a constant number of active bins available to accept items at any point during processing. The performance of the algorithm is measured by comparing the produced packing with an optimal offline packing of the list L into bins of size 1. The competitive ratio then becomes a function of the on-line bin size b. Csirik and Woeginger studied this problem in [J. Csirik, G.J. Woeginger, Resource augmentation for online bounded space bin packing, Journal of Algorithms 44(2) (2002) 308-320] and proved that no on-line bounded space algorithm can perform better than a certain bound ρ(b) in the worst case. We relax the on-line condition by allowing a complete repacking within the active bins, and show that the same lower bound holds for this problem as well, and repacking may only allow one to obtain the exact best possible competitive ratio of ρ(b) having a constant number of active bins, instead of achieving this bound in the limit. We design a polynomial time on-line algorithm that uses three active bins and achieves the exact best possible competitive ratio ρ(b) for the given problem.     

ON-LINE PROBLEMS OF MINIMIZING MAKESPAN ON A SINGLE BATCH PROCESSING MACHINE WITH NONIDENTICAL JOB SIZES

The on-line problem of scheduling on a batch processing machine with nonidentical job sizes to minimize makespan is considered. The batch processing machine can process a number of jobs simultaneously as long as the total size of these jobs being processed does not exceed the machine capacity. The processing time of a batch is given by the longest processing time of any job in the batch. Each job becomes available at its arrival time, which is unknown in advance, and its processing time becomes known upon its arrival. The paper deals with two variants： the case only with two distinct arrival times and the general case. For the first case, an on-line algorithm with competitive ratio 119/44 is given. For the latter one, a simple algorithm with competitive ratio 3 is given. For both variants the better ratios can be obtained if the problem satisfies proportional assumption.     

Optimal on-line flow time with resource augmentation

We study the problem of scheduling n jobs that arrive over time. We consider a non-preemptive setting on a single machine. The goal is to minimize the total flow time. We use extra resource competitive analysis: an optimal off-line algorithm which schedules jobs on a single machine is compared to a more powerful on-line algorithm that has ? machines. We design an algorithm of competitive ratio , where Δ is the maximum ratio between two job sizes, and provide a lower bound which shows that the algorithm is optimal up to a constant factor for any constant ?. The algorithm works for a hard version of the problem where the sizes of the smallest and the largest jobs are not known in advance, only Δ and n are known. This gives a trade-off between the resource augmentation and the competitive ratio.We also consider scheduling on parallel identical machines. In this case the optimal off-line algorithm has m machines and the on-line algorithm has ?m machines. We give a lower bound for this case. Next, we give lower bounds for algorithms using resource augmentation on the speed. Finally, we consider scheduling with hard deadlines, and scheduling so as to minimize the total completion time.     

Minimizing the total completion time on-line on a single machine, using restarts

We give an algorithm to minimize the total completion time on-line on a single machine, using restarts, with a competitive ratio of 3/2. The optimal competitive ratio without using restarts is 2 for deterministic algorithms and e/(e−1)≈1.582 for randomized algorithms. This is the first restarting algorithm to minimize the total completion time that is proved to be better than an algorithm that does not restart.     

平行机上带有前瞻区间的不相容工件组在线排序问题

研究当不相容工件组的个数与机器数相等时,具有前瞻区间的单位工件平行机无界平行分批在线排序问题.工件按时在线到达, 目标是最小化 最大完工时间. 具有前瞻区间是指在时刻t, 在线算法能预见到时间区间(t,t+\beta) 内到达的所有工件的信息.不可相容的工件组是指属于不同组的工件不能被安排在同一批中加工. \beta\geq 1 时, 提供了一个最优的在线算法; 当0\leq \beta < 1时, 提供了一个竞争比为1+\alpha 的最好可能的在线算法, 其中\alpha是方程\alpha^{2}+(1+\beta) \alpha+\beta-1=0的一个正根.最后, 给出了当\beta =0 时稠密算法竞争比的下界,并提供了达到该下界的最好可能的稠密算法.     

On-line k-Truck Problem and Its Competitive Algorithms

In this paper, based on the Position Maintaining Strategy (PMS for short), on-line scheduling of k-truck problem, which is a generalization of the famous k-server problem, is originally presented by our team. We proposed several competitive algorithms applicable under different conditions for solving the on-line k-truck problem. First, a competitive algorithm with competitive ratio 2k+1/ is given for any 1. Following that, if (c+1)/(c-1) holds, then there must exist a (2k-1)-competitive algorithm for k-truck problem, where c is the competitive ratio of the on-line algorithm about the relevant k-server problem. And then a greedy algorithm with competitive ratio 1+/, where lambda is a parameter related to the structure property of a given graph, is given. Finally, competitive algorithms with ratios 1+1/ are given for two special families of graphs.     

Worst-case analysis of a dynamic channel assignment strategy

We consider the problem of channel assignment in cellular networks with arbitrary reuse distance. We show upper and lower bounds for the competitive ratio of a previously proposed and widely studied version of dynamic channel assignment, which we refer to as the greedy algorithm. We study two versions of this algorithm: one that performs reassignment of channels, and one that never reassigns channels to calls. For reuse distance 2, we show tight bounds on the competitive ratio of both versions of the algorithm. For reuse distance 3, we show non-trivial lower bounds for both versions of the algorithm.     

A class of on-line scheduling algorithms to minimize total completion time

We consider the problem of scheduling jobs on-line on a single machine and on identical machines with the objective to minimize total completion time. We assume that the jobs arrive over time. We give a general 2-competitive algorithm for the single machine problem. The algorithm is based on delaying the release time of the jobs, i.e., making the jobs artificially later available to the on-line scheduler than the actual release times. Our algorithm includes two known algorithms for this problem that apply delay of release times. The proposed algorithm is interesting since it gives the on-line scheduler a whole range of choices for the delays, each of which leading to 2-competitiveness.We also show that the algorithm is 2α competitive for the problem on identical machines where α is the performance ratio of the Shortest Remaining Processing Time first rule for the preemptive relaxation of the problem.     

The on-line asymmetric traveling salesman problem

We consider two on-line versions of the asymmetric traveling salesman problem with triangle inequality. For the homing version, in which the salesman is required to return in the city where it started from, we give a $\frac{3+\sqrt{5}}{2}$-competitive algorithm and prove that this is best possible. For the nomadic version, the on-line analogue of the shortest asymmetric Hamiltonian path problem, we show that the competitive ratio of any on-line algorithm depends on the amount of asymmetry of the space in which the salesman moves. We also give bounds on the competitive ratio of on-line algorithms that are zealous, that is, in which the salesman cannot stay idle when some city can be served.     

On-line scheduling on a batch machine to minimize makespan with limited restarts

We study the on-line scheduling on an unbounded batch machine to minimize makespan. In this model, jobs arrive over time and batches are allowed limited restarts. Any batch that contains a job which has already been restarted once cannot be restarted any more. We provide a best possible on-line algorithm for the problem with a competitive ratio .     

工件按加工长度不增序到达的最小化最大流程在线分批排序

研究单处理机工件按加工长度不增顺序到达的在线分批排序问题．工件按时在线到达,目标是最小化最大流程．流程时间是指工件的完工时间与到达时间的差值,它体现了工件在系统内的逗留时间．对于批容量有界的情形,给出了一个竞争比为1＋√5/2的最好可能的在线算法;对于批容量无界的情形,给出了一个竞争比为√2的最好可能的在线算法．     

可拒绝的同类机在线排序(英文)

考虑了带拒绝费用的在线同类机排序模型.工件一个一个的到达,到达后或被接受,或以一定的费用被拒绝,目标是最小化最大完工时间与总的拒绝费用之和.我们提供了一个在线算法和分析了算法的竞赛比.     

Scheduling with step-improving processing times

We consider the scheduling problem of minimizing the makespan on a single machine with step-improving job processing times around a common critical date. For this problem we give an NP-hardness proof, a fast pseudo-polynomial time algorithm, an FPTAS, and an on-line algorithm with best possible competitive ratio.     

On-line scheduling with non-crossing constraints

We consider the problem of on-line scheduling with non-crossing constraints. The objective is to minimize the latest completion time. We provide optimal competitive ratio heuristics for the on-line list and on-line time problems with unit processing times, and a 3-competitive heuristic for the general on-line time problem.     

On-line and first fit colorings of graphs

A graph coloring algorithm that immediately colors the vertices taken from a list without looking ahead or changing colors already assigned is called “on-line coloring.” The properties of on-line colorings are investigated in several classes of graphs. In many cases we find on-line colorings that use no more colors than some function of the largest clique size of the graph. We show that the first fit on-line coloring has an absolute performance ratio of two for the complement of chordal graphs. We prove an upper bound for the performance ratio of the first fit coloring on interval graphs. It is also shown that there are simple families resisting any on-line algorithm: no on-line algorithm can color all trees by a bounded number of colors.