Given a 2SAT instance in CNF where each clause has at most two literals. Let $m$ be the number of clauses and $n$ be the number of variables et let $k$ be a positive number.

Question: Is there a truth assignment such that the number of variables that are true is $k$ or more?

I cannot find the name of this problem, if it was already studied? If so, is it NP-hard?

  • $\begingroup$ You are probably interested in a satisfying assignment. $\endgroup$ – Yuval Filmus Oct 17 '19 at 20:17

Your problem is known as Weighted-2-satisfiability, and is known to be NP-complete. The easiest way to see that is by reduction from Vertex Cover (exercise).

Note that the link above is about whether there is a satisfying assignment having at most $k$ true variables. This is equivalent to your problem (exercise).


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