Integer linear programs of configurations, or configuration IPs, are a classical tool in the design of algorithms for scheduling and packing problems where a set of items has to be placed in multiple target locations. Herein, a configuration describes a possible placement on one of the target locations, and the IP is used to choose suitable configurations covering the items. We give an augmented IP formulation, which we call the module configuration IP. It can be described within the framework of n-fold integer programming and, therefore, be solved efficiently. As an application, we consider scheduling problems with setup times in which a set of jobs has to be scheduled on a set of identical machines with the objective of minimizing the makespan. For instance, we investigate the case that jobs can be split and scheduled on multiple machines. However, before a part of a job can be processed, an uninterrupted setup depending on the job has to be paid. For both of the variants that jobs can be executed in parallel or not, we obtain an efficient polynomial time approximation scheme (EPTAS) of running time $$f(1/\varepsilon )\cdot \mathrm {poly}(|I|)$$ f ( 1 / ε ) · poly ( | I | ) . Previously, only constant factor approximations of 5/3 and $$4/3 + \varepsilon $$ 4 / 3 + ε , respectively, were known. Furthermore, we present an EPTAS for a problem where classes of (non-splittable) jobs are given, and a setup has to be paid for each class of jobs being executed on one machine.
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...
We consider two related scheduling problems: single resource-constrained scheduling on identical parallel machines and a generalization with resource-dependent processing times. In both problems, jobs require a certain amount of an additional resource and have to be scheduled on machines minimizing the makespan, while at every point in time a given resource capacity is not exceeded. In the first variant of the problem, the processing times and resource amounts are fixed, while in the second the former depends on the latter. Both problems contain bin packing with cardinality constraint as a special case, and, therefore, these problems are strongly NP-complete even for a constant number of machines larger than three, which can be proven by a reduction from 3-Partition. Furthermore, if the number of machines is part of the input, then we cannot hope for an approximation algorithm with absolute approximation ratio smaller than 3/2. We present asymptotic fully polynomial time approximation schemes (AFPTAS) for the problems: For any ε > 0, a schedule of length at most (1+ε) times the optimum plus an additive term of O ( p max log (1/ε)/ε) is provided, and the running time is polynomially bounded in 1/ε and the input length. Up to now, only approximation algorithms with absolute approximation ratios were known. Furthermore, the AFPTAS for resource-constrained scheduling on identical parallel machines directly improves the additive term of the best AFPTAS for bin packing with cardinality constraint so far.
We consider a special case of the scheduling problem on unrelated machines, namely the Restricted Assignment Problem with two different processing times. We show that the configuration LP has an integrality gap of at most 5 3 ≈ 1.667 for this problem. This allows us to estimate the optimal makespan within a factor of 5 3 , improving upon the previously best known estimation algorithm with ratio 11 6 ≈ 1.833 due to Chakrabarty, Khanna, and Li [2].
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