Ronald L. Graham

Optimization and Approximation in Deterministic Sequencing and Scheduling: a Survey

The theory of deterministic sequencing and scheduling has expanded rapidly during the past years. In this paper we survey the state of the art with respect to optimization and approximation algorithms and interpret these in terms of computational complexity theory. Special cases considered are single machine scheduling, identical, uniform and unrelated parallel machine scheduling, and open shop, flow shop and job shop scheduling. We indicate some problems for future research and include a selective bibliography.

An efficient algorith for determining the convex hull of a finite planar set

Step Find a point Pin the plane w%ch is in &he In&

Worst-Case Performance Bounds for Simple One-Dimensional Packing Algorithms

x of Cl-l(s). At worst, this can be done in clfl sQp9 by te dting 3 element subsets of S for collineti@, discarding middle p&n& of collinear rets ar5b bq@rig when the fust noncollinear set (if there i# or&j. &y X, y arrcf z, is found. P can be chosen to I&E the centroid oC the triangle formed by X, y and z. Sfq 2: Express each si E S in polar coordinates th origin P and 8 = 0 in the direction of zu~ arhitnry fixed half-line L from P. This canversion can be done in c2n operations ior some rimed constant

Optimal scheduling for two-processor systems

The following abstract problem models several practical problems in computer science and operations research: given a list L of real numbers between 0 and l, place the elements of L into a minimum number

On the History of the Minimum Spanning Tree Problem

L^ *

The Complexity of Computing Steiner Minimal Trees

of “bins” so that no bin contains numbers whose sum exceeds l. Motivated by the likelihood that an excessive amount of computation will be required by any algorithm which actually determines an optimal placement, we examine the performance of a number of simple algorithms which obtain “good” placements. The first-fit algorithm places each number, in succession, into the first bin in which it fits. The best-fit algorithm places each number, in succession, into the most nearly full bin in which it fits. We show that neither the first-fit nor the best-fit algorithm will ever use more than

The steiner problem in phylogeny is NP-complete

\frac{17}{10}L^ * + 2

Quasi-random graphs

bins. Furthermore, we outline a proof that, if L is in decreasing order, then neither algorithm will use more than

Some NP-complete geometric problems

\frac{11}{9} L^ * + 4

COMPLEXITY RESULTS FOR BANDWIDTH MINIMIZATION

bins. Examples are given to show that both upper bou...