The equivalence of two classical list scheduling algorithms for dependent typed tasks with release dates, due dates and precedence delays

Abstract

We consider a finite set of unit time execution tasks with release dates, due dates and precedence delays. The machines are partitionned into k classes. Each task requires one machine from a fixed class to be executed. The problem is the existence of a feasible schedule. This general problem is known to be N P-complete; many studies were devoted to the determination of polynomial time algorithms for some special subcasses, most of them based on a particular list schedule. The Garey-Johnson and Leung-Palem-Pnueli algorithms (respectively GJ and LPP in short) are both improving the due dates to build a priority list. They are modifiying them using necessary conditions until a fix point is reached. The present paper shows that these two algorithms are different implementations of a same generic one. The main consequence is that all the results valid for GJ algorithm, are also for LPP and vice versa. Keywords list scheduling algorithms • polynomial sub-problems • approximation algorithms 1 Introduction Scheduling problems with release and due dates have been considered for a long time, either in their decision version (is there a schedule meeting all the constraints ?) or in their optimization version, by minimizing the maximum lateness L max. Most of the decision problems are N P-complete once precedence and resource constraints are considered [6,12,15]. However, some particular instances have led to polynomial algorithms. Garey and Johnson [5] solved polynomially P 2|prec, p i = 1, r i , d i |L max and its preemptive version P 2|prec, pmtn, r i , d i |L max by using a particular list scheduling algorithm. This algorithm was extended to get approximation algorithms for the L max criteria with various hypothesis: parallel processors [8], preemptive jobs [7], communication delays [7], typed tasks systems, and unitary resource constraints scheduling problems [1].

Structural Properties of an Open Problem in Preemptive Scheduling

“…Gabow [11] designed an almost linear-time algorithm for the minimum-makespan problem. Leung, Palem, and Pnueli [15] and Carlier, Hanen, and Munier-Kordon [5] extend these results to precedence delays. Baptiste and Timkovsky [4] focus on minimization of total completion time and present an O(n 9 ) time shortest-path optimization algorithm for scheduling jobs with release dates.…”

“…We show that ξ j +1 (b) = 0, which will make our first transformation in (5) feasible. This holds for j + 1 = τ(a), since by definition of A-configuration b J(ξ τ(a) ).…”