TWO APPROXIMATION SCHEMES FOR SCHEDULING ON PARALLEL MACHINES UNDER A GRADE OF SERVICE PROVISION

2012 ◽  
Vol 29 (05) ◽  
pp. 1250029 ◽  
Author(s):  
WEIDONG LI ◽  
JIANPING LI ◽  
TONGQUAN ZHANG
Keyword(s):  
Polynomial Time ◽  
Parallel Machines ◽  
Approximation Scheme ◽  
Approximation Schemes ◽  
Polynomial Time Approximation Scheme ◽  
Time Approximation ◽  
Running Time ◽  
Grade Of Service ◽  
Special Case ◽  
Polynomial Time Approximation

We consider the offline scheduling problem to minimize the makespan on m parallel and identical machines with certain features. Each job and machine are labeled with the grade of service (GoS) levels, and each job can only be executed on the machine whose GoS level is no more than that of the job. In this paper, we present an efficient polynomial-time approximation scheme (EPTAS) with running time O(n log n) for the special case where the GoS level is either 1 or 2. This partially solves an open problem posed in (Ou et al., 2008). We also present a simpler full polynomial-time approximation scheme (FPTAS) with running time O(n) for the case where the number of machines is fixed.

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

2009 ◽  
Vol 01 (03) ◽  
pp. 355-368 ◽  
Author(s):  
JIANPING LI ◽  
WEIDONG LI ◽  
JIANBO LI
Keyword(s):  
Polynomial Time ◽  
Service Provision ◽  
Processing Time ◽  
Approximation Scheme ◽  
Allocation Problem ◽  
Approximation Schemes ◽  
Polynomial Time Approximation Scheme ◽  
Time Approximation ◽  
Running Time ◽  
Grade Of Service

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 > 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.

FULLY POLYNOMIAL-TIME APPROXIMATION SCHEMES FOR THE MAX–MIN CONNECTED PARTITION PROBLEM ON INTERVAL GRAPHS

2012 ◽  
Vol 04 (01) ◽  
pp. 1250005 ◽  
Author(s):  
BANG YE WU
Keyword(s):  
Polynomial Time ◽  
Approximation Scheme ◽  
Interval Graph ◽  
Interval Graphs ◽  
Approximation Schemes ◽  
Partition Problem ◽  
Polynomial Time Approximation Scheme ◽  
Time Approximation ◽  
Connected Subgraphs ◽  
Polynomial Time Approximation

We study how to partition an interval graph with non-negative vertex weights into k connected subgraphs such that the minimum total weight of any part of the partition is maximized. For k = 2, it is shown that for any ε > 0, a (1 + ε)-approximation can be found in O((1/ε)n3) time, i.e., it admits a fully polynomial-time approximation scheme (FPTAS). For any fixed k > 2, the problem also admits an FPTAS when restricted to k-connected interval graphs.

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