Asymptotically optimal algorithms for job shop scheduling and packet routing

We propose asymptotically optimal algorithms for the job shop scheduling an d packet routing problems. We propose a fluid relaxation for the job shop s cheduling problem in which we replace discrete jobs with the flow of a cont inuous fluid. We compute an optimal solution of the fluid relaxation in clo sed form, obtain a lower bound C-max to the job shop scheduling problem, an d construct a feasible schedule from the fluid relaxation with objective va lue at most C-max + O(root C-max) where the constant in the O(.) notation i s independent of the number of jobs, but it depends on the processing time of the jobs, thus producing an asymptotically optimal schedule as the total number of jobs tends to infinity. If the initially present jobs increase p roportionally, then our algorithm produces a schedule with value at most C- max + O(1). For the packet routing problem with fixed paths the previous al gorithm applies directly. For the general packet routing problem we propose a linear programming relaxation that provides a lower bound C-max and an a symptotically optimal algorithm that uses the optimal solution of the relax ation with objective value at most C-max + O(root C-max). Unlike asymptotic ally optimal algorithms that rely on probabilistic assumptions, our propose d algorithms make no probabilistic assumptions and they are asymptotically optimal for all instances with a large number of jobs (packets). In computa tional experiments our algorithms produce schedules which are within 1% of optimality even for moderately sized problems. (C) 1999 Academic Press.