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Set cover for $d=2$ is the edge cover problem, which is a poly-time problem [1]. For $d=3$, it is indeed NP-complete. This can be shown using the same reduction from 3SAT to 3-dimensional matching shown in [2], in which given a 3SAT instance, one can construct a universe $U=\{1,\dots,N\}$ and a set $S$ of 3-sets such that there is an exact cover of $U$ (i.e ...

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