Polynomial time reducibility, NP-hard and NP-complete:

We add up some formal definitions to the common concept of problem reduction.

Definition: The function f: A* -> A* is a polynomial-time computable if and only if there is a polynomial p and a DTM M that, when started with w on its tape, halts with the f(w) on its tape, and tM( w ) ≤ p( |w|).

Definition: L is a polynomial-time reducible to L’, represented by L ≤p L’ if and only if there is polynomial-time computable f: A* -> A* in such a way that ∀w ∈ A*, w ∈ L if and only if f(w) ∈ L’.

In another words, the question w ∈ L? Can be answered by deciding the f(w) ∈ L’. Therefore, the complexity of deciding membership in L is utmost the complexity of computing f plus the complexity of deciding the membership in L’.

As we generously avoid polynomial times, this justifies the notation L ≤p L’.

The remarkable fact makes the theory of NP fascinating and rich. With respect to class NP and the notion of polynomial-time reducibility, there are ‘toughest problems’ in the sense that all the decision problems in NP are reducible to any one of such ‘toughest problems’.

Definition: L’ is NP-hard if and only if ∀L ∈ NP, L ≤p L’
Definition: L’ is NP-complete if and only if L’ ∈ NP and L’ is NP-hard

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