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Even computing a maximum independent set of unit axis-parallel squares is known to be np-hard: 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 $...


NP-hardness is proved by Roth and Viswanathan in the paper On the hardness of decoding the gale-berlekamp code

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