Tight Approximations for Resource Constrained

We consider the following resource constrained scheduling problem. Given m iden-of unit length, where each job T j has a start time r j 2 II N, requires one processor and an amount R i (j) 2 f0; 1g of resource R i , i = 1; : : :; s. The optimization problem is to schedule the jobs at discrete times in II 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 to schedule 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 RR ock 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, Graham, Johnson and Yao (1976), gave for every > 0 a linear time algorithm with an asymptotic approximation guarantee of (s +)C opt. The main contributions of this paper include the rst polynomial-time algorithm approximating C opt for every 2 (0; 1) within a factor of 1 + for instances with b i = (?2 log(Cs)) for all i and m = (?2 logC), and a proof that the achieved approximation under the given condition is best possible, unless P = NP. Furthermore, in some cases for every xed > 1 a parallel 2-factor approximation algorithm can be derived.