Np-complete

NP-Complete refers to the hardest known problems within the complexity class NP

Np-hard

NP-hard problems (Non-deterministic Polynomial-time hard problems) are those problems which are not easier than any problem in NP; in other words, an algorithm for an NP-hard problem can be used to solve any problem in NP by transforming the input in polynomial time

Others

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If there exists a np-hard problem that is not in np to the best of my knowledge no such problem has been proved to fall in this category at this moment of time such problem is from question NP-Complete VS NP-Hard As i understand it an np-hard problem is not from question What are the differences between NP, NP-Complete and NP-Hard? |

Unfortunately there is little hope to find an algorithm which is much better than brute-force considering that the problem is actually np-hard but not even np-complete;a proof of np-hardness of this problem is that the minimum vertex cover problem well known to be np-hard and not np-complete from question Dependency Algorithm - find a minimum set of packages to install |

The decision variant asks whether there is such a subset of size k and is np-complete making the optimization variant np-hard;the sets here are always exactly 4 elements which is still np-hard from question Optimized solution to find distinct squares of 2X2 in a matrix of 0s and 1s |