A note on the complexity of task scheduling with communication contention

Considering the contention for communication resources in task scheduling is discussed in [3]. Theorem 1 of [3] states the NP-completeness of the associated decision problem and the corresponding proof shows NP-completeness in the weak sense [2]. Here, a new proof is presented for Theorem 1 of [3] that shows NP-completeness in the strong sense. The proof is based on a reduction from the well-known 3-PARTITION [2] problem, which is NP-complete in the strong sense. This document is meant to be read in conjunction with [3]. Theorem 1 (NP-Completeness – Contention Model): Let G = (V,E, w, c) be a DAG and MTG = ((N,P,D,H, b), ω) a parallel system. The decision problem C-SCHED(G, MTG), associated with the scheduling problem is as follows: Is there a schedule S for G on MTG with length sl(S) ≤ T, T ∈ Q? C-SCHED(G, MTG) is NP-complete. Proof. First, it is argued that C-SCHED belongs to NP, then it is shown that C-SCHED is NP-hard by reducing the well-known NP-complete problem 3-PARTITION [2] in polynomial time to C-SCHED. 3-PARTITION is NP-complete in the strong sense. The 3-PARTITION problem is: Given a set A of 3m positive integer numbers ai and a positive integer bound B such that ∑ 3m i=1 ai = mB with B/4 < ai < B/2 for i = 1 . . . 3m. Can A be partitioned into m disjoint sets A1, . . .,Am (triplets) such that each Ai, i = 1 . . .m, contains exactly 3 elements of A, whose sum is B? Pierre-François Dutot is with ID-IMAG, Grenoble, France; Email: [email protected] Oliver Sinnen is with Dept. of Electrical and Computer Engineering, University of Auckland, Private Bag 92019, Auckland, New Zealand; Email: [email protected] Leonel Sousa, Senior Member, IEEE, is with INESC-ID, Instituto Superior Técnico, Technical University of Lisbon, Rua Alves Redol 9, P-1000-029 Lisboa, Portugal; Email: [email protected]