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I have two 2D arrays: $P[n][s]$ and $C[n][s]$, $s \leq n$.

P contains sets of nodes and $C$ the cost of a set in $P$, e.g. the cost of $P[2][2]$ is $C[2][2]$ and a set $p \in P = \{ s_0, s_1, ..., s_{n-1} \}^q, q \leq n$, e.g. $p=\{s_0, s_3\}$.

For every row I can pick at most one set but in the end the union of the picked sets needs to contain all nodes $\{s_0, s_1, ..., s_{n-1} \}$.

How do I find the least expensive picks in linear time?

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If you are able to find the least expensive picks in linear (or even polynomial) time then $P=NP$.

Your problem is a generalization of the set-cover problem: given a set $X$ of items and a collection $S = \{S_1, \dots, S_m \} \subseteq 2^X$ of subsets of $X$, find $S' \subseteq S$ such that $\cup_{S_i \in S'} S_i = X$ and $|S'|$ is minimized.

This is exactly your problem once you set $n=|X|$, $s=1$, $P[i][1] = S_i$, and $C[i][1]=1$.

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  • $\begingroup$ Well, that's a pity. I was hoping to solve my earlier problem easily ->cs.stackexchange.com/questions/127227/… $\endgroup$ – rn42v1r Jun 18 '20 at 20:31
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    $\begingroup$ Yeah, I noticed that this problem was also generalizing your earlier question :) $\endgroup$ – Steven Jun 18 '20 at 20:33

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