All-norm approximation algorithms

Abstract

A major drawback in optimization problems and in particular in scheduling problems is that for every measure there may be a different optimal solution. In many cases the various measures are different ℓ p norms. We address this problem by introducing the concept of an All-norm ρ-approximation algorithm, which supplies one solution that guarantees ρ-approximation to all ℓ p norms simultaneously. Specifically, we consider the problem of scheduling in the restricted assignment model, where there are m machines and n jobs, each is associated with a subset of the machines and should be assigned to one of them. Previous work considered approximation algorithms for each norm separately. Lenstra et al.[11] showed a 2-approximation algorithm for the problem with respect to the ℓ ∞ norm. For any fixed ℓ p norm the previously known approximation algorithm has a performance of θ(p). We provide an all-norm 2-approximation polynomial algorithm for the restricted assignment problem. On the other hand, we show that for any given ℓ p norm (p > 1) there is no PTAS unless P=NP by showing an APXhardness result. We also show for any given ℓ p norm a FPTAS for any fixed number of machines.

Optimal location with equitable loads

“…The processing time of job i on machine j is w i for j ∈ J , and ∞ otherwise. This special scheduling model is called the restricted assignment model (Azar et al 2004). Lenstra et al (1990) gave a 2-approximation algorithm for this problem even when p is variable.…”

“…Lenstra et al (1990) gave a 2-approximation algorithm for this problem even when p is variable. A stronger result which produces a feasible solution which is at most 2 times the optimal solution for all l q norms (q ≥ 1) simultaneously, is presented in Azar et al (2004). (In the l q -norm model, the goal is to minimize ((δ 1 ) q + · · · + (δ p ) q ) 1/q , the l q norm of the vector (δ 1 , .…”

Scheduling for energy minimization on restricted parallel processors

“…Finally, we prove that our algorithm ECSEMRPP_Algo (see Algorithm 5) solves the SEMRPP problem optimally for the case of uniform tasks by finding the min-max load vector ⃗ l, which is a strongly optimal assignment defined in [2,5].…”

Structural properties of time-dependent scheduling problems with the l norm objective

“…Bansal and Pruhs [4] considered the problem of minimizing the average quality of service in server systems, applying the l p norm as a tool for balancing average quality. Azar et al [3] described application of the l p norm to modeling the work of a disk, where the weight of a job corresponds to the disk access frequency. Since a delay of the disk is proportional to the load on the machine it is assigned to, the average delay can be measured using the l p norm with p = 2.…”

“…Since a delay of the disk is proportional to the load on the machine it is assigned to, the average delay can be measured using the l p norm with p = 2. The same authors [3] and Caragiannis [6] considered scheduling problems on unrelated machines with the l p norm. Gupta and Sivakumar [16] applied a weighted l p norm as an optimality criterion in a single-machine multi-objective scheduling problem.…”