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The Class P

[Section 7.2]. The Class P. Def 7.12: The class P consists of languages that are decidable in polynomial time, i.e., P = [ k TIME(n k ) Example: PATH = { <G,s,t> | G is a digraph that has a path from s to t }. [Section 7.2]. The Class P.

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The Class P

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  1. [Section 7.2] The Class P Def 7.12: The class P consists of languages that are decidable in polynomial time, i.e., P = [k TIME(nk) Example: PATH = { <G,s,t> | G is a digraph that has a path from s to t }

  2. [Section 7.2] The Class P Def 7.12: The class P consists of languages that are decidable in polynomial time, i.e., P = [k TIME(nk) Example: RELPRIME = { <x,y> | x and y are relatively primes }

  3. [Section 7.2] The Class P Def 7.12: The class P consists of languages that are decidable in polynomial time, i.e., P = [k TIME(nk) Example: PRIME = { <x> | x is a prime }

  4. [Section 7.3] The Class NP Example: HAMPATH = { <G,s,t> | G is digraphs with Hamiltonian path from s to t }

  5. [Section 7.3] The Class NP Example: COMPOSITES = { x | x=pq, for some p,q > 1 }

  6. [Section 7.3] The Class NP Def 7.18: A verifier for a language A is an algorithm A, where A = { w | V accepts <w,c> for some string c } Polynomial-time verifier runs in (deterministic) time polynomial in the length of w. The string c is called the certificate, or proof, of the membership in A.

  7. [Section 7.3] The Class NP Def 7.18: A verifier for a language A is an algorithm A, where A = { w | V accepts <w,c> for some string c } Polynomial-time verifier runs in (deterministic) time polynomial in the length of w. The string c is called the certificate, or proof, of the membership in A. Def 7.19:NP is the class of languages that have polynomial time verifiers.

  8. [Section 7.3] The Class NP Thm 7.20: A language is in NP iff it is decided by some nondeterministic polynomial-time TM. Def 7.21: NTIME(t(n)) = { L | L is a language decided by a O(t(n))-time nondeterministic TM }. Thus, NP = [k NTIME(nk)

  9. [Section 7.3] The Class NP Thm 7.20: A language is in NP iff it is decided by some nondeterministic polynomial-time TM. Def 7.21: NTIME(t(n)) = { L | L is a language decided by a O(t(n))-time nondeterministic TM }. Thus, NP = [k NTIME(nk)

  10. [Section 7.3] The Class NP Example: CLIQUE = { <G,k> | G is undirected graph with a k-clique }

  11. [Section 7.3] The Class NP Example: SAT = { <Á> | Á is a satisfiable Boolean formula }

  12. [Section 7.3] The Class NP • Wrapping up: • P – exists polynomial-time algorithm • NP – exits polynomial-time verifier • BIG open problem: • Is P = NP ??? • Note: also exists a class coNP, the class of complements of problems in NP (e.g. CLIQUEC, “is every clique of a given graph of different size than k?”). We do not know if NP = coNP.

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