Sartaj Sahni

P-Complete Approximation Problems

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.

Open Shop Scheduling to Minimize Finish Time

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.

Algorithms for Scheduling Independent Tasks

The following job sequencing problems are studied: (i) single processor job sequencing with deadlines, (ii) job sequencing on m-identical processors to minimize finish time and related problems, (iii) job sequencing on 2-identical processors to minimize weighted mean flow time. Dynamic programming type algorithms are presented to obtain optimal solutions to these problems, and three general techniques are presented to obtain approximate solutions for optimization problems solvable in this way. The techniques are applied to the problems above to obtain polynomial time algorithms that generate “good” approximate solutions.

Computing Partitions with Applications to the Knapsack Problem

Given <italic>r</italic> numbers <italic>s</italic><subscrpt>1</subscrpt>, ···, <italic>s<subscrpt>r</subscrpt></italic>, algorithms are investigated for finding all possible combinations of these numbers which sum to <italic>M</italic>. 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 for Scheduling Nonidentical Processors

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 algorithms 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 O(n log mn).

Parallel Matrix and Graph Algorithms

Matrix multiplication algorithms for cube connected and perfect shuffle computers are presented. It is shown that in both these models two

Flowshop and Jobshop Schedules: Complexity and Approximation

n \times n

Data broadcasting in SIMD computers

matrices can be multiplied in

Approximate Algorithms for the 0/1 Knapsack Problem

O(n/m + \log m)

Computationally Related Problems

time when