# what is the LP gap of this particular non metric facility location problem in planar graphs?

Suppose I have a facility location problem with all service costs 0 (or infinity (in particular service costs are not metric)) such that the edges $$ij$$ for which facility $$i$$ can service client $$j$$ with cost 0 (together with the facilities and clients) forms a planar graph.

What is the maximum integrality gap of such an LP?

This is exactly set cover with the additional constraint that the set element incidence graph ( that is a graph with nodes $$v_S$$ for every set $$S$$ and nodes $$v_e$$ for every element $$e$$ and the edges $$v_S v_e$$ for elements $$e$$ and sets $$S$$ for which $$e \in S$$) is planar.

What is the LP gap if the graph is not planar but say fixed genus/minor free etc?

Also, does this problem have a name like planar/bounded genus set cover or something?