Abstract
This paper presents a scheduling problem with coupled-tasks in presence of a compatibility graph on a mono processor. We investigate a specific configuration, in which the coupled-tasks possess an idle time equal to 2. The complexity of these problems will be studied according to the presence or absence of triangles in the compatibility graph. As an extended matching, we propose a polynomial-time algorithm which consists in minimizing the number of non-covered vertices, by covering vertices with edges or paths of length two in the compatibility graph. This type of covering will be denoted 2-cover technique. According on the compatibility graph type, the 2-cover technique provides an 13/12-approximation or 10/9-approximation algorithm.