Approximation schemes for scheduling on parallel machines

Abstract

We discuss scheduling problems with m identical machines and n jobs where each job has to be assigned to some machine. The goal is to optimize objective functions that solely depend on the machine completion times.As a main result, we identify some conditions on the objective function, under which the resulting scheduling problems possess a polynomial time approximation scheme. Our result contains, generalizes, improves, simplifies, and unifies many other results in this area in a natural way.

Power-efficient assignment of virtual machines to physical machines

“…Hence, their results can be directly applied in the (·, m)-VMA problem. Later, Epstein et al [20] extended [19] further for the uniformly related machines case. We will use these results in Section 3 in the analysis of the offline case of (·, m)-VMA and (·, ·)-VMA.…”

“…Similarly, Alon et al [19] extended this work for a more general set of functions, that include f (·) as defined above. Hence, their results can be directly applied in the (·, m)-VMA problem.…”

Power-efficient assignment of virtual machines to physical machines

“…Hence, their results can be directly applied in the (·, m)-VMA problem. Later, Epstein et al [20] extended [19] further for the uniformly related machines case. We will use these results in Section 3 in the analysis of the offline case of (·, m)-VMA and (·, ·)-VMA.…”

“…Similarly, Alon et al [19] extended this work for a more general set of functions, that include f (·) as defined above. Hence, their results can be directly applied in the (·, m)-VMA problem.…”

Searching for Multiple Mobile Users

“…u1 has a search deficit of 7/8, and therefore, at the first paging cycle, a search is conducted for u1 only at c1, while the search for u2 and u3 is conducted at four locations for each, with a success probability of 19 24 each. The expected number of searches for u1, u2, and u3 is 3 1 2 locations for u1 and 3 3 4 locations each for u2 and u3. The expected total search cost is therefore 11 paging channels, which is significantly less than the cost of 15 3 16 searches expected from [14].…”

On the optimality of exact and approximation algorithms for scheduling problems

“…Given our results, it seems that a 2 1/ log O(1) (1/ ) + n O (1) time PTAS is possible, and it is indeed the case under certain conditions. Jansen and Robenek [18] give a 2 O(1/ log 2 (1/ )) + n O (1) time PTAS under a certain conjecture, and Chen et al [6] provide a 2 O(1/ log 2 (1/ )) + O(n) time PTAS if every machine can accept only a constant number of jobs (the lower bound also holds true if we restrict that each machine can accept at most c jobs for c ≥ 4).…”

Power-efficient assignment of virtual machines to physical machines

“…Hence, their results can be directly applied in the (·, m)-VMA problem. Later, Epstein et al [20] extended [19] further for the uniformly related machines case. We will use these results in Section 3 in the analysis of the offline case of (·, m)-VMA and (·, ·)-VMA.…”

“…Similarly, Alon et al [19] extended this work for a more general set of functions, that include f (·) as defined above. Hence, their results can be directly applied in the (·, m)-VMA problem.…”

Searching for Multiple Mobile Users

“…u1 has a search deficit of 7/8, and therefore, at the first paging cycle, a search is conducted for u1 only at c1, while the search for u2 and u3 is conducted at four locations for each, with a success probability of 19 24 each. The expected number of searches for u1, u2, and u3 is 3 1 2 locations for u1 and 3 3 4 locations each for u2 and u3. The expected total search cost is therefore 11 paging channels, which is significantly less than the cost of 15 3 16 searches expected from [14].…”

On the optimality of exact and approximation algorithms for scheduling problems

“…Given our results, it seems that a 2 1/ log O(1) (1/ ) + n O (1) time PTAS is possible, and it is indeed the case under certain conditions. Jansen and Robenek [18] give a 2 O(1/ log 2 (1/ )) + n O (1) time PTAS under a certain conjecture, and Chen et al [6] provide a 2 O(1/ log 2 (1/ )) + O(n) time PTAS if every machine can accept only a constant number of jobs (the lower bound also holds true if we restrict that each machine can accept at most c jobs for c ≥ 4).…”

Power-efficient assignment of virtual machines to physical machines

“…Hence, their results can be directly applied in the (·, m)-VMA problem. Later, Epstein et al [20] extended [19] further for the uniformly related machines case. We will use these results in Section 3 in the analysis of the offline case of (·, m)-VMA and (·, ·)-VMA.…”

“…Similarly, Alon et al [19] extended this work for a more general set of functions, that include f (·) as defined above. Hence, their results can be directly applied in the (·, m)-VMA problem.…”

Searching for Multiple Mobile Users

“…u1 has a search deficit of 7/8, and therefore, at the first paging cycle, a search is conducted for u1 only at c1, while the search for u2 and u3 is conducted at four locations for each, with a success probability of 19 24 each. The expected number of searches for u1, u2, and u3 is 3 1 2 locations for u1 and 3 3 4 locations each for u2 and u3. The expected total search cost is therefore 11 paging channels, which is significantly less than the cost of 15 3 16 searches expected from [14].…”

On the optimality of exact and approximation algorithms for scheduling problems

“…Given our results, it seems that a 2 1/ log O(1) (1/ ) + n O (1) time PTAS is possible, and it is indeed the case under certain conditions. Jansen and Robenek [18] give a 2 O(1/ log 2 (1/ )) + n O (1) time PTAS under a certain conjecture, and Chen et al [6] provide a 2 O(1/ log 2 (1/ )) + O(n) time PTAS if every machine can accept only a constant number of jobs (the lower bound also holds true if we restrict that each machine can accept at most c jobs for c ≥ 4).…”