  A set family ${\cal F}$ is $uncrossable$ if $A \cap B,A \cup B \in {\cal F}$
or $A \setminus B,B \setminus A \in {\cal F}$ for any $A,B \in {\cal F}$. A
classic result of Williamson, Goemans, Mihail, and Vazirani [STOC 1993:708-717]
states that the problem of covering an uncrossable set family by a min-cost
edge set admits approximation ratio $2$, by a primal-dual algorithm. They asked
whether this result extends to a larger class of set families and combinatorial
optimization problems. We define a new class of $semi$-$uncrossable$ $set$
$families$, when for any $A,B \in {\cal F}$ we have that $A \cap B \in {\cal
F}$ and one of $A \cup B,A \setminus B ,B \setminus A$ is in ${\cal F}$, or $A
\setminus B,B \setminus A \in {\cal F}$. We will show that the Williamson et
al. algorithm extends to this new class of families and identify several
``non-uncrossable'' algorithmic problems that belong to this class. In
particular, we will show that the union of an uncrossable family and a monotone
family, or of an uncrossable family that has the disjointness property and a
proper family, is a semi-uncrossable family, that in general is not
uncrossable. For example, our result implies approximation ratio $2$ for the
problem of finding a min-cost subgraph $H$ such that $H$ contains a Steiner
forest and every connected component of $H$ contains at least $k$ nodes from a
given set $T$ of terminals.
