Bicriteria graph theory problem? What do you call this?

Hi all,
I've been thinking about a certain class of problems that I haven't
read about in literature (I guess "bicriteria graph theory" is the
closest I can think of). I'm interested in solving graph theory
problems when we have two constraints to follow.

For example (I doubt this is easily solvable or even approximated),
suppose we have a multigraph with edges doubly weighted by "cost" and
"demand". I want a subset of edges such that for every node, the sum
of the demands of all the edges incident to this node in our subset is
greater than or equal to 1, while the sum of the costs of all the edges
in the subset is minimized.

Has anyone heard of problems like this before? I know of a
diameter-constrained steiner tree problem that featured a bicriteria
construction, but I'd be curious to know if anyone's seen something
like this.

Thanks!
KH

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