An EPTAS for Scheduling on Unrelated Machines of Few Different Types

Abstract

In the classical problem of scheduling on unrelated parallel machines, a set of jobs has to be assigned to a set of machines. The jobs have a processing time depending on the machine and the goal is to minimize the makespan, that is the maximum machine load. It is well known that this problem is NP-hard and does not allow polynomial time approximation algorithms with approximation guarantees smaller than 1.5 unless P=NP. We consider the case that there are only a constant number K of machine types. Two machines have the same type if all jobs have the same processing time for them. This variant of the problem is strongly NP-hard already for K = 1. We present an efficient polynomial time approximation scheme (EPTAS) for the problem, that is, for any ε > 0 an assignment with makespan of length at most (1 + ε) times the optimum can be found in polynomial time in the input length and the exponent is independent of 1/ε. In particular we achieve a running time of 2 O(K log(K) 1 /ε log 4 1 /ε) + poly(|I|), where |I| denotes the input length. Furthermore, we study three other problem variants and present an EPTAS for each of them: The Santa Claus problem, where the minimum machine load has to be maximized; the case of scheduling on unrelated parallel machines with a constant number of uniform types, where machines of the same type behave like uniformly related machines; and the multidimensional vector scheduling variant of the problem where both the dimension and the number of machine types are constant. For the Santa Claus problem we achieve the same running time. The results are achieved, using mixed integer linear programming and rounding techniques. * This work was partially supported by the German Research Foundation (DFG) project JA 612/16-1. The current article is an extended version of the conference article [20] arXiv:1701.03263v2 [cs.DS] 6 Dec 2017 Basic Concepts. We study polynomial time approximation algorithms: Given an instance I of an optimization problem, an α-approximation A for this problem produces a solution in time poly(|I|), where |I| denotes the input length. For the objective function value A(I) of this solution it is guaranteed that A(I) ≤ αOPT(I), in the case of an minimization problem, or A(I) ≥ (1/α)OPT(I), in the case of an maximization problem, where OPT(I) is the value of an optimal solution. We call α the approximation guarantee or rate of the algorithm. In some cases a polynomial time approximation scheme (PTAS) can be achieved, that is, an (1+ε)-approximation for each ε > 0. If for such a family of algorithms the running time can be bounded by f (1/ε)poly(|I|) for some computable function f , the PTAS is called efficient (EPTAS), and if the running time is polynomial in both 1/ε and |I| it is called fully polynomial (FPTAS). Related Work.It is well known that the unrelated scheduling problem admits an FPTAS in the case that the number of machines is considered constant [16] and we already mentioned the seminal work by Lenstra et al. [23]. Furthermore, the problem of unrelated scheduling w...

Approximation Results for Makespan Minimization with Budgeted Uncertainty

“…Recall also that C Γ (σ i ) = p(σ i ) +p Γ (σ i ), where p(σ i ) = j∈σ i p j , and p Γ (σ i ) is the sum of thep j values of the Γ largest jobs (w.r.t.p j ) of σ i (or the sum of all p j values if |σ i | ≤ Γ ). To obtain a PTAS for P |U Γ |C max , we will reduce to the following problem, which admits an EPTAS (see [8]).…”

“…Notice that the EPTAS of [8] for this problem provides an (1 + ǫ)-approximation running in time f (|I|, ǫ, k) = 2 O(k log(k) 1 ǫ log 4 ( 1 ǫ )) + poly(|I|). We also introduce the following decision problem.…”

A Unified Framework for Designing EPTAS’s for Load Balancing on Parallel Machines

“…Machines with pre-specified type, ψ = 1 and π j = ∞ ∀j EPTAS [17] Machines with pre-specified type, ψ = 0, s i = 1 ∀i, and π j = ∞ ∀j PTAS [3] τ = 2, α i (1) = 0 ∀i, p j (1) = ∞ ∀j, ψ = 1, and π j = ∞ ∀j PTAS [19] Table 1. Summary of previous studies of special cases of problem P .…”

High Multiplicity Scheduling on Uniform Machines in FPT-Time

“…In the context of approximation, Jansen, Klein, and Verschae [13] gave an EPTAS (FPT (1 + ε)-approximation algorithm with parameter 1 ε ) for Q||C max and Woeginger [26] gave a PTAS (XP (1 − ε)-approximation algorithm with parameter 1 ε ) for P ||C min . Jansen and Maack [14] gave an approximation scheme for R||{C min , C max } that runs in time O * (f (τ, ε)) (where τ is the number of different machine types) for some function f ; they also gave an approximation scheme for Q||C max that runs in time O * (g(ε)) for some function g. Bansal and Sviridenko [1] gave an O( log log m log log log m )-approximation algorithm for P |p ij ∈ {p j , ∞}|C min . For the objective C envy , there cannot be a (multiplicative) approximation algorithm as it is NP-hard to decide if there is a schedule for that assigns the same load to every machine, even for scheduling on identical machines [2].…”