In this paper,a two-stage semi-hybrid flowshop problem which appears in graphics processing is studied. For this problem, there are two machines M1 and M2, and a set of independent jobs J= {J1 ,J2 ,…,Jn }. Each Ji consists of two tasks Ai and Bi ,and task Ai must be completed before task Bi can start. Furthermore ,task Ai can be processed on M1 for ai time units ,or on Mw for ai^J time units ,while task Bi can only be processed on M2 for bi time units. Jobs and machines are available at time zero and no preemption is allowed. The objective is to minimize the maximum job completion time. It is showed that this problem is NP-hard. And a pseudo-polynomial time optimal algorithm is presented. A polynomial time approximation algorithm with worst-case ratio 2 is also presented.
In this paper, we consider the seml-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.
This work is aimed at investigating the online scheduling problem on two parallel and identical machines with a new feature that service requests from various customers are entitled to many different grade of service (GoS) levels, so each job and machine are labelled with the GoS levels, and each job can be processed by a particular machine only when its GoS level is no less than that of the machine. The goal is to minimize the makespan. For non-preemptive version, we propose an optimal online al-gorithm with competitive ratio 5/3. For preemptive version, we propose an optimal online algorithm with competitive ratio 3/2.
In this paper,a new concept of an optimal complete multipartite decomposition of type 1 (type 2) of a complete n-partite graph Q n is proposed and another new concept of a normal complete multipartite decomposition of K n is introduced.It is showed that an optimal complete multipartite decomposition of type 1 of K n is a normal complete multipartite decomposition.As for any complete multipartite decomposition of K n,there is a derived complete multipartite decomposition for Q n.It is also showed that any optimal complete multipartite decomposition of type 1 of Q n is a derived decomposition of an optimal complete multipartite decomposition of type 1 of K n.Besides,some structural properties of an optimal complete multipartite decomposition of type 1 of K n are given.
Huang QingxueDept. of Math., Zhejiang Univ., Hangzhou 310027, China.
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 （2m-3）/（m-1） for any m〉2 and present optimal semi-online algorithms for m = 2, 3.
