Scheduling stretched coupled-tasks with compatibilities constraints : model, complexity and approximation results for some class of graphs
We tackle the makespan minimization coupled-tasks problem in presence of compatibility constraints. In particular, we focus on stretched coupled-tasks, {\it i.e.}coupled-tasks having the same sub-tasks execution time and idle time duration. We study severals problems in frame works of classic complexity and approximation for which the compatibility graph $G_c$ is bipartite (star, chain, $\ldots$) In such context, we design some efficient polynomial-time approximation algorithms according to difference parameters of the scheduling problem. When $G_c$ is a $k$-stage bipartite graph, we propose, among other, a $\frac{7}{6}$-approximation algorithm when $k=1$, and a $\frac{13}{9}$-approximation algorithm when $k=2$.\\