I have an index taking as keys values from the power set $P(S)$ of a set $S$, except for $\emptyset$ and $S$.
Then I have a query $Q=(s, k)$, where $s \in P(S) - \{\emptyset \cup S\}$ and $ 1 < k \le |S|$.
The result of the query is the set of covers $R$ of $S$; $\forall r \in R, |r| \le k$.
So for instance, if $S = \{a, b, c, d\}$ and $Q=( \{a,b\}, 3 )$, $R$ should return all covers of $S$ formed by $\{a, b\}$ and $k = (3 - 1 = ) 2$ or less other subsets of $P(S) - \{\emptyset \cup S\}$, namely: $R = \{$ $$ \{a,b\} , \{c\} , \{d\}$$ $$ \{a,b\} , \{c, d\} $$ $$ \{a,b\} , \{a, c, d\} $$ $$ \{a,b\} , \{b, c, d\} $$ $\}$
I want to know if there is an efficient algorithm that can give me all of these combinations of covers.
In other words, I want to know if there is an efficient way to get all covers formed by at most $k$ elements, where I only know 1 element out of k.