Question

Prove that polynomial-time reducibility is transitive: that is, if L1is polynomial-time-reducible to L2, and L2 is polynomial-time-reducible to L3, then L1 is polynomial-time-reducible to L3.

Answer 1

Answer:-------------

If L1 ≤P L2 and L2 ≤P L3 then L1 ≤P L3

Proof.----------
If f is a polynomial time reduction from L₁ to L₂ running in time n^k and g is a polynomial time reduction from L₂ to L₃ computed in time n^m then g ◦ f is a reduction from L₁ to L₃
and can be computed in time O(n^k + (n^k)^m ) = O(n^km).

Hence Proved.