NP-Completeness

NP-Complete problems are the hardest problems (no problem is harder) in NP, i.e., every problem in NP reduces to an NP-Complete problem.

• A language L $\subseteq$ {0,1}* is NP-Complete if L $\in$ NP, and for every L' $\in$ NP.
• The class of NP-Complete languages is called NPC.
• A language L $\subseteq$ {0,1}* is NP-Hard if for every L' $\in$ NP.
• A language that is NP-Hard is not necessarily in NP.
  E.g., Kth Largest Subset is NP-Hard, but not NPC.
  KLS: are there at least K distinct subsets A' of set A such that $\sum_{a \in A'} a \leq B$?

If we can solve one NPC problem in polynomial time, we can solve every problem in NP in polynomial time. For this reason, many assume P $\neq$ NP.