Teofilo F. Gonzalez

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.

CLUSTERING TO MINIMIZE THE MAXIMUM INTERCLUSTER DISTANCE

The problem of clustering a set of points so as to minimize the maximum intercluster distance is studied. An O(kn) approximation algorithm, where n is the number of points and k is the number of clusters, that guarantees solutions with an objective function value within two times the optimal solution value is presented. This approximation algorithm succeeds as long as the set of points satisfies the triangular inequality. We also show that our approximation algorithm is best possible, with respect to the approximation bound, if PZ NP.

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.

Flowshop and Jobshop Schedules: Complexity and Approximation

We show that finding minimum finish time preemptive and non-preemptive schedules for flow shops and job shops is NP-complete. Bounds on the performance of various heuristics to generate reasonably good schedules are also obtained.

BOUNDS FOR LPT SCHEDULES ON UNIFORM PROCESSORS

We study the performance of LPT (largest processing time) schedules with respect to optimal schedules in a nonpreemptive multiprocessor environment. The processors are assumed to have different speeds and the tasks being scheduled are independent.

Preemptive Scheduling of Uniform Processor Systems

AaSTRACT An O(n) t~me algorithm is presented to obtain an opt,mal fimsh time preemptive schedule for n independent tasks on m uniform processors This algorithm assumes that the tasks are lnmally ordered by task length and that the umform processors are ordered by processor speed

Covering a set of points in multidimensional space

Abstract Let P = { p 1 , p 2 ,…, p n } be a set of points in d -space. We study the problem of covering with the minimum number of fixed-size orthogonal hypersquares ( CS d for short) all points in P . We present a fast approximation algorithm that generates provably good solutions and an improved polynomial-time approximation scheme for this problem. A variation of the CS d problem is the CR d problem, covering by fixed-size orthogonal hyperrectangles, where the covering of the points is by hyperrectangles with dimensions D 1 , D 2 ,…, D d instead of hypersquares of size D . Another variation is the CD d problem, where we cover the set of points with hyperdiscs of diameter D . Our algorithms can be easily adapted to these problems.

Unit Execution Time Shop Problems

The problem of preemptively and nonpreemptively scheduling a set of n independent jobs on an m machine open shop, flow shop or job shop is studied. It is shown that the problem of constructing optimal mean finishing time preemptive and nonpreemptive schedules is NP-hard. These problems are not only NP-hard in the strong sense, but remain NP-hard even when all nonzero tasks have identical execution time requirements. These results will also apply to the case when the problem is to construct an optimal finish time preemptive and nonpreemptive schedule for a flow shop or a job shop. We also discuss the problem of constructing no-wait schedules for these problems.

Stochastic local search algorithms for multiobjective combinatorial optimization: a review

The information provided is the sole responsibility of the authors and does not necessarily reflect the opinion of the members of IRIDIA. The authors take full responsability for any copyright breaches that may result from publication of this paper in the IRIDIA – Technical Report Series. IRIDIA is not responsible for any use that might be made of data appearing in this publication.

A New Algorithm for Preemptive Scheduling of Trees

An algorithm which schedules forests of <italic>n</italic> tasks on <italic>m</italic> identical processors in <italic>O</italic>(<italic>n</italic> log <italic>m</italic>) time, offline, is given. The schedules are optimal with respect to finish time and contain at most <italic>n</italic> - 2 preemptions, a bound which is realized for all <italic>n</italic>. Also given is a simpler algorithm which runs in <italic>O</italic>(<italic>nm</italic>) time on the same problem and can be adapted to give optimal finish time schedules on-line for independent tasks with release times.