We study some classical problems in scheduling. In the rst problem a sequence of jobs must be scheduled on m identical parallel machines. As each job arrives, its processing time(pj) is known. We have to assign the job to one of the machines immediately, without knowledge of the jobs that arrive later. Jobs cannot be preempted. The goal is to minimize the makespan, the time when the last job is completed. The second problem deals with scheduling jobs on a single machine/processor. With each job we have an associated weight(wj), release time(rj) and processing time(pj). Now the objective is to minimize the weighted sum of completion times,Pnj=0wjCj . The constraints are that a job can only be scheduled after its release time, and preemption is not allowed. In the third problem again there are m machines. Each job has to visit the machines in the same xed order. The processing time pij of job j on machine i is known. We want a schedule which minimizes the makespan. A job can only be processed on a single machine at a time. Chapter

OR Scheduling Algorithms

In this paper, we prove that there exists a schedule for routing any set of packets with edge-simple paths, on any network, inO(c+d) steps, wherec is the congestion of the paths in the network, andd is the length of the longest path. The result has applications to packet routing in parallel machines, network emulations, and job-shop scheduling.

Packet routing and job-shop scheduling in O (congestion+dilation) steps

In the job shop scheduling problem, there are $m$ machines and $n$ jobs. A job consists of a sequence of operations, each of which must be processed on a specified machine, and the aim is to complete all jobs as quickly as possible. This problem is strongly ${\cal NP}$-hard even for very restrictive special cases. The authors give the first randomized and deterministic polynomial-time algorithms that yield polylogarithmic approximations to the optimal length schedule. These algorithms also extend to the more general case where a job is given not by a linear ordering of the machines on which it must be processed but by an arbitrary partial order. Comparable bounds can also be obtained when there are $m'$ types of machines, a specified number of machines of each type, and each operation must be processed on one of the machines of a specified type, as well as for the problem of scheduling unrelated parallel machines subject to chain precedence constraints.

Improved approximation algorithms for shop scheduling problems

We consider the following general correlation-clustering problem [1]: given a graph with real edge weights (both positive and negative), partition the vertices into clusters to minimize the total absolute weight of cut positive edges and uncut negative edges Thus, large positive weights (representing strong correlations between endpoints) encourage those endpoints to belong to a common cluster; large negative weights encourage the endpoints to belong to different clusters; and weights with small absolute value represent little information In contrast to most clustering problems, correlation clustering specifies neither the desired number of clusters nor a distance threshold for clustering; both of these parameters are effectively chosen to be the best possible by the problem definition

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Correlation Clustering with Partial Information

Shop scheduling problems are notorious for their intractability, both in theory and practice. In this paper, we construct a polynomial approximation scheme for the flow shop scheduling problem with an arbitrary fixed number of machines. For the three common shop models (open, flow, and job), this result is the only known approximation scheme. Since none of the three models can be approximated arbitrarily closely in the general case (unless P = NP), the result demonstrates the approximability gap between the models in which the number of machines is fixed, and those in which it is part of the input of the instance. The result can be extended to flow shops with job release dates and delivery times and to flow shops with a fixed number of stages, where the number of machines at any stage is fixed. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

S. V. Sevast'yanov

Papers

Bounding algorithm for the routing problem with arbitrary paths and alternative servers