# Is Geometric Disjoint Set Cover in P?

I have come across the following optimisation subproblem:

Geometric Disjoint Set Cover. Consider a collection $$C$$ of (not necessarily distinct) ranges taken from a universe range $$X \subset \mathbb{R}^d$$. Does there exist a $$k$$-partition of $$C$$ into set covers of $$X$$?

I am particularly interested in the case where ranges are axis-parallel rectangles in $$\mathbb{R}^2$$, and $$k$$ is small. Does there exist a polynomial time algorithm for fixed $$k$$ and $$d$$? Is there a more widely known name I can search by?

Without the geometric constraint, this problem is NP-hard even for $$k = 2$$, by equivalence to deciding $$2$$-colourability for hypergraphs. On the other hand, when $$d = 1$$ there is a simple greedy $$O(|C| \log |C|)$$ algorithm that works for all $$k$$: at each step consume up to the $$k$$-th rightmost endpoint of intervals starting at or before the current position. Can this be generalised to $$d = 2$$?