  We consider the problem of assigning tasks to agents under time conflicts,
with applications also to frequency allocations in point-to-point wireless
networks. In particular, we are given a set $V$ of $n$ agents, a set $E$ of $m$
tasks, and $k$ different time slots. Each task can be carried out in one of the
$k$ predefined time slots, and can be represented by the subset $e\subseteq E$
of the involved agents. Since each agent cannot participate to more than one
task simultaneously, we must find an allocation that assigns non-overlapping
tasks to each time slot. Being the number of slots limited by $k$, in general
it is not possible to executed all the possible tasks, and our aim is to
determine a solution maximizing the overall social welfare, that is the number
of executed tasks. We focus on the restriction of this problem in which the
number of time slots is fixed to be $k=2$, and each task is performed by
exactly two agents, that is $|e|=2$. In fact, even under this assumptions, the
problem is still challenging, as it remains computationally difficult. We
provide parameterized complexity results with respect to several reasonable
parameters, showing for the different cases that the problem is fixed-parameter
tractable or it is paraNP-hard.
