  Obtaining accepting lassos, witnesses and winning strategies in
omega-automata and games with omega-regular winning conditions is an integral
part of many formal methods commonly found in practice today. Despite the fact
that in most applications, the lassos, witnesses and strategies found should be
as small as possible, little is known about the hardness of obtaining small
such certificates. In this paper, we survey the known hardness results and
complete the complexity landscape for the cases not considered in the
literature so far. We pay particular attention to the approximation hardness of
the problems as approximate small solutions usually suffice in practice.
