I'm solving Problem 14.4 of What can be computed?.

14.4 Define the decision problem SAT-5CLAUSES as follows. The input is a Boolean formula B in CNF. The solution is “yes” if it is possible to simultaneously satisfy at least 5 of B’s clauses. Is SAT-5CLAUSES NP-complete? Give a rigorous proof of your answer.

Is there any suggestion to solve this problem?


1 Answer 1


It is unknown whether the problem is $NP$-complete. In particular, you problem is non-trivial and it is in $P$. Therefore it is $NP$-complete if and only if $P=NP$.

To see that your problem is in $P$ you can consider the following "brute-force" algorithm. Let $n$ be the number of variables and $n$ be the number of clauses.

If it is possible to simultaneously satisfy $5$ clauses, then there are $5$ literals that are true and belong to different clauses.

You can then explicitly consider all subsets of $\min\{n, 5\}$ variables and, for each subset $S$, try all possible (constantly many) truth assignments to the variables in $S$.

For each of these $O(n^5)$ (partial) truth assignments, check whether at least $5$ clauses are satisfied. This can be done in constant time after a linear-time preprocessing of the input instance.

The overall time required is then $O(mn + n^5)$ where $O(mn)$ is an upper bound on the size of the instance.


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