Complexity and Approximation for the Precedence Constrained Scheduling Problem with Large Communication Delays

General scheduling non-approximability results in presence of hierarchical communications

“…Indeed, in [16], the authors proved that there is no possibility of finding a q-approximation with q < 1 + 1/(c + 4) (unless P ¼ NP) for the problem P jprec; c ij ¼ c P 2; p i ¼ 1jC max and for C max = c + 1 the problem is polynomial, and for C max = c + 2 the problem is partially open. For C max = c + 3 the complexity of this problem is unknown.…”

“…Recently, in [16], the authors proved that there is no possibility of finding a q-approximation with q < 1 + 1/(c + 4) (unless P ¼ NP) for the case where all tasks of the precedence graph have unit execution times, where the multiprocessor is composed of an unrestricted number of machines, and where c denotes the communication delay between two tasks i and j both submitted to a precedence constraint and which have to be processed by two different machines (this problem is denoted in the following UET-LCT (Unit Execution Time Large Communication Time) homogeneous scheduling communication delays problem). The problem becomes polynomial whenever the makespan is at most (c + 1).…”

Multiprocessor Scheduling

“…Figure 1.4. A partial precedence graph for the NT1 -completeness of the scheduling problem , cij = c 3, pi = Theorem 1.3.1 T/ze problem of deciding whether an instance of , cij = c ; pi = has a schedule of length equal or less than (c+4) is -complete with c 3 (see (Giroudeau et al, 2005)). Proof It is easy to see that , cij = c ; pi = = c + 4 .…”

“…It easy to see that there is a schedule of length equal or less than (c + 4) if only if there is a truth assignment such that each clause in has exactly one true literal (i.e. one literal equal to 1), see (Giroudeau et al, 2005). For the special case c = , by using another polynomial-time trnasformation, we state: Theorem 1.3.2 The problem of deciding whether an instance of , cij = 2; pi = has a schedule of length equal or less than six is -complete (see (Giroudeau et al, 2005)).…”

Scheduling in Bandwidth Constrained Tree Networks

“…Addressing these scheduling issues is not new. Besides a large amount of work being done in practical scheduling literature, there has been some theoretical work done offline with the goal of minimizing makespan [8,9,16,17,20,21]. Additionally, over two decades ago Phillps, Stein and Wein [32] introduced an interesting model of network scheduling and argued its importance in practice.…”