Suppose a holding owns some companies.  Each company produces a set of
products. Each product is produced by at most two companies.  As an
example, pasta is produced by Barilla and Saiwa, while wine is produced
only by Barilla \footnote{foodnote: Any coincidence with existing
companies is purely casual.}. 

Suppose the holding experiences a crisis and has to sell one company.  The
holding's policy is to keep on producing all products.  This clearly makes
it impossible to sell some companies --as an example the Barilla company
in the above situation, because it would be impossible to produce wine. 
Anyway the managers are even more cautious:  They know that in the future
it may be necessary to sell more companies, and they do not want to get
into a situation in which they will not be able to produce all products. 
More formally, they are interested in the minimal sets of companies that
produce all products.  A company is {\it strategic\/} if it is in at least
one of such minimal sets.  Therefore a query which is very relevant to the
managers is whether a company is strategic or not: They prefer to sell a
non-strategic company first, because after the transaction the minimal
sets of companies that produce all products remain the same. 


Now let us consider a slightly more complex situation, in which up to
three (say) companies can {\it control\/} another company.  As an example,
companies Barilla and Saiwa together have control over company Frutto. 

Assume now that the following constraint is imposed for determining
strategic co mpanies: A company $A$ which is controlled by the companies
$B_1$, \ldots, $B_i$ can be sold only if also one of the companies $B_1$,
\ldots, $B_i$ is sold. This constraint completely changes the minimal sets
of companies that produce all pro ducts. 

In the former case --no controlled companies-- the problem of deciding
whether a company is strategic is in the complexity class NP (cf.\
\cite{cado-lenz-94}), while in the latter case the same problem is easily
shown to be complete for the class $\Sigma^p_2$ of the polynomial
hierarchy (cf.\ \cite{eite-gott-93a}). 

