For P-complete problems such as traveling salesperson, cycle covers, 0-1 integer programming, multicommodity network flows, quadratic assignment, etc., it is shown that the approximation problem is also P-complete. In contrast with these results, a linear time approximation algorithm for the clustering problem is presented.

A linear time algorithm to obtain a minimum finish time schedule for the two-processor open shop together with a polynomial time algorithm to obtain a minimum finish time preemptive schedule for open shops with more than two processors are obtained. It is also shown that the problem of obtaining minimum finish time nonpreemptive schedules when the open shop has more than two processors is NP-complete.

Given r numbers s 1 , …, s r , algorithms are investigated for finding all possible combinations of these numbers which sum to M . This problem is a particular instance of the 0-1 unidimensional knapsack problem. All of the usual algorithms for this problem are investigated in terms of both asymptotic computing times and storage requirements, as well as average computing times. We develop a technique which improves all of the dynamic programming methods by a square root factor. Empirical studies indicate this new algorithm to be generally superior to all previously known algorithms. We then show how this improvement can be incorporated into the more general 0-1 knapsack problem obtaining a square root improvement in the asymptotic behavior. A new branch and search algorithm that is significantly faster than the Greenberg and Hegerich algorithm is also presented. The results of extensive empirical studies comparing these knapsack algorithms are given

Exact and approximate algorithms are presented for scheduling independent tasks in a multiprocessor environment in which the processors have different speeds. Dynamic programming type algorithms are presented which minimize finish time and weighted mean flow time on two processors. The generalization to m processors is direct. These algorithms have a worst-case complexity which is exponential in the number of tasks. Therefore approximation algorithms of low polynomial complexity are also obtained for the above problems. These algomthms are guaranteed to obtain solutions that are close to the optimal. For the case of minimizing mean flow time on m-processors an algorithm is given whose complexity is 0(n log ran).KEY WORDS AND PHRASES: scheduling independent tasks, uniform processors, unrelated processors, finish time, mean flow time, weighted mean flow time, exact algorithms, approximate algorithm, complexity CR CATEGORIES: 4.32, 5.39 ] ntroductionWe are concerned here with scheduling n >_ 1 independent tasks Tl, ..-, T, on nn >_ 1 processors P~, • • • , P~. Thus we assume there are no precedence constraints on the tasks and also that all schedules must be nonpreemptive. The execution time of task % on processor P, will be denoted by t, yielding an m X n matrix of processing times. Each t, is assumed to be a positive rational number and without loss of generality tie, 1 < j _~ n is normalized to a positive integer.Formally a schedule S for m processors is a partition of the set of task indices, { 1, 2, • • • , n I into m disjoint, ordered sets R1, • • • , R,~ such that (i) R, = {r,l,r~2, "." , r,~,}, j, > 0, (ii) U~<,_