NPcomplete

NPcomplete - Harvard CS 121 and CSCI E-207 Lecture 21...

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon
Harvard CS 121 and CSCI E-207 Lecture 21: NP-Completeness Harry Lewis November 24, 2009 Reading: Sipser § 7.4, § 7.5. For “culture”: Computers and Intractability: A Guide to the Theory of NP-completeness , by Garey & Johnson.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Harvard CS 121 November 24, 2009 Boolean logic Boolean formulas Def : A Boolean formula (B.F.) is either: · a “Boolean variable” x,y,z,. .. · ( α β ) where α,β are B.F.’s. · ( α β ) where α,β are B.F.’s. · ¬ α where α is a B.F. e.g. ( x y z ) ( ¬ x ∨ ¬ y ∨ ¬ z ) [Omitting redundant parentheses] 1
Background image of page 2
Harvard CS 121 November 24, 2009 Boolean satisfiability Def: A truth-assignment is a mapping A : Boolean variables → { 0 , 1 } . [ 0 = false, 1 = true] A T-A is extended to all B.F.’s by the rules: · A ( α β ) = 1 iff A ( α ) = 1 or A ( β ) = 1 · A ( α β ) = 1 iff A ( α ) = 1 and A ( β ) = 1 · A ( ¬ α ) = 1 iff A ( α ) = 0 A satisfies α (written A | = α ) iff A ( α ) = 1 . In this case, α is satisfiable . If no A satisfies α , then α is unsatisfiable . SAT = { α : α is a satisfiable Boolean formula } . Prop: SAT NP 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Harvard CS 121 November 24, 2009 A “similar” problem in P: 2-SAT A 2-CNF formula is one that looks like ( x y ) ( ¬ y z ) ( ¬ y ∨ ¬ x ) i.e., a conjunction of clauses , each of which is the disjunction of 2 literals (or 1 literal, since ( x ) ( x x ) ) 2-SAT = the set of satisfiable 2-CNF formulas. e.g. ( x y ) ( ¬ x ∨ ¬ y ) ( ¬ x y ) ( x ∨ ¬ y ) / SAT 3
Background image of page 4
Harvard CS 121 November 24, 2009 2-SAT P Method (resolution): 1. If x and ¬ x are both clauses, then not satisfiable e.g. ( x ) ( z y ) ( ¬ x ) 2. If ( x y ) ( ¬ y z ) are both clauses, add clause ( x z ) (which is implied). 3. Repeat. If no contradiction emerges satisfiable. O ( n 2 ) repetitions of step 2 since only 2 literals/clause. 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Harvard CS 121 November 24, 2009 P vs. NP We would like to solve problems in NP efficiently. We know P NP. Problems in P can be solved “fairly” quickly. What is the relationship between P and NP? 5
Background image of page 6
Harvard CS 121 November 24, 2009 NP and Exponential Time Claim : NP S k TIME (2 n k ) Of course, this gets us nowhere near P. Is P = NP? i.e., do all the NP problems have polynomial time algorithms? It doesn’t “feel” that way but as of today there is no NP problem that has been proven to require exponential time! 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
November 24, 2009 The Strange, Strange World if P = NP Thousands of important languages can be decided in polynomial time, e.g. S
Background image of page 8
Image of page 9
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 31

NPcomplete - Harvard CS 121 and CSCI E-207 Lecture 21...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online