Extended matching problem for a coupled-tasks scheduling problem

This paper introduces a scheduling problem with coupledtasks in presence of a compatibility graph on a single 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 polynomialtime 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 by 2-cover technique. According on the compatibility graph type, the 2-cover technique provides a polynomial-time ρ-approximation algorithm with ρ = 13 12 (resp. ρ = 10 9 ) in absence (resp. presence) of triangles.