Tight approximations for resource constrained scheduling and bin packing

• Published in: Discrete Applied Mathematics Vol. 79; no. 1; pp. 223 - 245
• Main Authors: Srivastav, Anand, Stangier, Peter
• Format: Journal Article
• Language: English
• Published: Lausanne Elsevier B.V 27.11.1997; Amsterdam Elsevier; New York, NY
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Approximation algorithm; Chromatic index; Combinatorial probability; Computer science; control theory; systems; Derandomization; Exact sciences and technology; Mathematical programming; Mathematics; Multidimensional bin packing; Operational research and scientific management; Operational research. Management science; Probability and statistics; Probability theory and stochastic processes; Randomized algorithm; Resource constrained scheduling; Scheduling, sequencing; Sciences and techniques of general use; Theoretical computing; 60E15; Multidimensional bin packing; 60C05; 68Q25; 90C10; 90B35; Randomized algorithm; Approximation algorithm; Resource constrained scheduling; Derandomization; Chromatic index; Optimization method; Bin packing problem; Linear programming; Scheduling; Algorithm; Computational complexity; Polynomial time
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/S0166-218X(97)00045-0

We consider the following resource constrained scheduling problem. We are given m identical processors, s resources R 1, …, R s with upper bounds b 1, …, b s , n independent jobs T 1, …, T n of unit length, where each job T j has a start time r j ϵ N, requires one processor and an amount R i ( j) ϵ {0, 1} of resource R i , i = 1, …, s. The optimization problem is to schedule the jobs at discrete times in N subject to the processor, resource and start-time constraints so that the latest scheduling time is minimum. Multidimensional bin packing is a special case of this problem. Resource constrained scheduling can be relaxed in a natural way when one allows the scheduling of fraction of jobs. Let C opt (resp. C) be the minimum schedule size for the integral (resp. fractional scheduling). While the computation of C opt is a NP-hard problem, C can be computed by linear programming in polynomial time. In case of zero start times Röck and Schmidt (1983) showed for the integral problem a polynomial-time approximation within ( (m 2 )C opt and de la Vega and Lueker (1981), improving a classical result of Garey et al. (1976), gave for every ε > 0 a linear time algorithm with an asymptotic approximation guarantee of ( s + ε) C opt . The main contributions of this paper include the first polynomial-time algorithm approximating C opt for every ε ϵ (0, 1) within a factor of 1 + ε for instances with b i = Ω(ε −2log(Cs)) for all i and m = Ω(ε −2log C) , and a proof that the achieved approximation under the given conditions is best possible, unless P = NP. Furthermore, in some cases we give for every fixed α > 1 a parallel 2α-factor approximation algorithm.