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Even computing a maximum independent set of unit axis-parallel squares is known to be np-hard: https://www.sciencedirect.com/science/article/pii/0020019081901113?via%3Dihub Since coloring is a "harder" problem, it should also be NP-hard. A constant approximation follows as if a point is covered by $k$ squares, then the chromatic number is at least \$...