Order 14: 211–228, 1997–1998. © 1998 Kluwer Academic Publishers. Printed in the Netherlands. 211 Chains and Trees: ‘Strong’–‘Weak’ Order in Job Scheduling MOSHE DROR MIS Department, College of Business, The University of Arizona, Tucson, AZ 85721, U.S.A. (Received: 5 February 1997; accepted: 15 December 1997) Abstract. We present a summary of recent NP-hardness and polynomial time solvability results for the distinction between ‘strong’ and ‘weak’ precedence for chains and trees in scheduling. We dis- tinguish between chains and proper trees which are not chains, and demonstrate that the strong-weak precedence distinction for chains is not inclusive with regards to NP-hardness, and conjecture that the same holds for strong-weak tree precedence. The objective is to show that different ‘interpretations’ for chain and tree order relations in scheduling might have far reaching computational implications. Mathematics Subject Classifications (1991): 05, 06, 68. Key words: partial order, scheduling, complexity. 1. Introduction Ordered sets have been of great importance both in many applied and theoretical problems in computer science and operations research, including ranging project management, processors scheduling, and many others. The basic structure of these problems requires partial orders such as precedence constraints in scheduling. For at least the last two decades it has been common knowledge within the computer science and combinatorial optimization community that most applied problems which involve partial orders are NP-hard. Still, since much attention has been paid to special classes of partial orders in search of “nice” structural properties which could lead to the design of efficient solution procedures, or at least to obtain bounds by structural relaxation, the delineation of the partial order in chain and tree structures into ‘strong’ and ‘weak’ precedence is of considerable interest. The ‘strong’–‘weak’ precedence distinction has clear interpretations for chain and tree precedence; this is not the case when series-parallel partial order is involved. Thus, we not pursue those concepts beyond the tree structure. This paper is a summary of the research results on the ‘strong’–‘weak’ prece- dence distinction presented in Dror et al. (1996a,b,c and 1997) written for the scheduling community, and its aim is to introduce those findings to a mathematical community interested in new concepts related to partial orders (precedence).