Grouping Techniques for Scheduling Problems: Simpler and Faster

Abstract

In this paper we describe a general grouping technique to devise faster and simpler approximation schemes for several scheduling problems. We illustrate the technique on two different scheduling problems: scheduling on unrelated parallel machines with costs and the job shop scheduling problem. The time complexity of the resulting approximation schemes is always linear in the number n of jobs, and the multiplicative constant hidden in the O(n) running time is reasonably small and independent of the error ε.

The load rebalancing problem

“…Given this value of OPT, we will obtain L T = 1, L E = 0, a 1 = 0, a 2 = 0, b 1 = 1, and b 2 = 0. It is easy to see that PARTITION will not make any moves whatsoever and will have performance ratio exactly 3 2 . 2…”

“…Corollary 1. The Constrained Load Rebalancing problem does not have a polynomial ρ-approximation algorithm, for any ρ < 3 2 , unless P = NP.…”

“…Theorem 6. The makespan minimization problem with costs c ij ∈ {p, q} (p < q) does not have a polynomial ρ-approximation algorithm, for any ρ < 3 2 , unless P = NP.…”

“…The best positive result known is the 2-approximation algorithm due to Shmoys and Tardos [15], via linear programming. When the number of processors is fixed, the generalized assignment problem admits a PTAS, as described in the work of Efraimidis and Spirakis [2], Jansen and Porkolab [9], and Fishkin, Jansen, and Mastrolilli [3].…”

A (3/2+ε) approximation algorithm for scheduling moldable and non-moldable parallel tasks

“…The LP for the large narrow jobs can be interpreted as a scheduling problem on a constant number O(1/δ 4 ) of unrelated processors [22]. This problem can be solved approximately with ratio (1 + α/2) in time O(n) + g(1/δ), where 1/α = poly(1/δ) and g is exponential in 1/δ [9]. Each approximate solution of the scheduling problem determines positions for large narrow jobs with load bounded by Π s,h +⌊αm⌋+αm/2 ≤ Π s,h +2⌊αm⌋ (using m ≥ 2/α).…”