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NP-Completeness NP-Completeness Limits of Computation Clayton Johnson Edna Reiter

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Limits of Computation by Johnson & Reiter 2 Two definitions for NP Definition 6.1 : A decision problem P is in NP if there is a Nondeterministic Turing Machine that runs in time bounded by a polynomial. Definition 6.2 : A decision problem is in NP if and only if it has a polynomial verifier Verifier = Turing machine that checks if a proposed solution C is truly a solution
Limits of Computation by Johnson & Reiter 3 Examples of verifiers Given a graph G, the sequence of nodes 1,4,2,9,8 (the certificate) is a path from node 1 to node 8 The boolean expression (x z)   (¬y z) (x ¬y) ( ¬y) is true for (x,y,z) = (1,0,1) = certificate Given an array A, the number M is the maximum value in the array. (M is the certificate) Given integer N and integers n 1 , …n m , there are k of them that sum to N

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Limits of Computation by Johnson & Reiter 4 Examples of verifiers Path 1,4,2,9,8: Check that there are edges 1-4, 4-2, 2- 9, and 9-8 Boolean: (x V z) (¬y V z)   (x V ¬y) ( ¬y) = (1 1) (1 1 ( 1 0)  ( 1 which is true A has max M: check a i <= M for all i in the array Verify each is polynomial!!!
Limits of Computation by Johnson & Reiter 5 Nondeterministic Poly-Time Theorem : A language L is in NP if and only if L can be decided by a poly-time nondeterministic TM; That is, definitions 6.1 and 6.2 are equivalent. Proof : Let A NP have an O(n k ) time verifier V. A polytime NTM can guess the O(n k ) certificate c for x A and check that it is a certificate. Let N be the O(n k ) time nondeterministic decider for B. The path to the accept state is a certificate, and is polynomial. The verifier only needs to show that this path does lead to the accept state.

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Limits of Computation by Johnson & Reiter 6 Nondeterministic Poly-Time Corollary : Let NTIME(t(n)) be the class of languages decidable by a O(t(n)) nondeterministic Turing machine. Then i ,... 2 , 1 k k ) n ( NTIME = = NP Aside: Most of the time, the certificate/verifier way of looking at NP problems gives more relevant information. Note that only proofs for x A are required, not for the complement x A.
Limits of Computation by Johnson & Reiter 7 Boolean Formulas and SAT A Boolean variable x can be TRUE or FALSE, which is also denoted by “1” or “0”. Standard Boolean operations are AND (x y), OR (x y), and NOT ( ¬ x or also x ). Typical Boolean formula : φ (x,y) = ( ¬ x y) (x ∨¬ y). This φ is satisfiable (by the assignments x=y=TRUE and x=y=FALSE). SAT = {< φ > | φ is a satisfiable Boolean formula}

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Limits of Computation by Johnson & Reiter 8 Conjunctive Normal Form Literal = Boolean variable (x) or its negation ( ¬ x). Clause = A set of OR-ed literals, like (x ¬ y z) A formula φ is in conjunctive normal form (CNF ), if it is the AND of clauses. For example: A 3CNF formula has only clauses with 3 literals: ) y y z ( ) z ( ) y x ( ) z , y , x ( = φ ) y y z ( ) z z z ( ) z y x ( ) z , y , x ( = φ 3SAT = {< φ > | φ is a satisfiable 3CNF formula}
Limits of Computation by Johnson & Reiter 9 A CNF Example

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