Last class Decision/Optimization 3-SAT  Independent-Set Independent-Set  3-SAT P, NP Cook’s Theorem NP-hard, NP-complete 3-SAT  Clique, Subset-Sum,

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Text of Last class Decision/Optimization 3-SAT  Independent-Set Independent-Set  3-SAT P, NP Cook’s...

  • Slide 1
  • Last class Decision/Optimization 3-SAT Independent-Set Independent-Set 3-SAT P, NP Cooks Theorem NP-hard, NP-complete 3-SAT Clique, Subset-Sum, 3-COL
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  • Reductions INSTANCE of A INSTANCE of B A B all reductions we had were: the black-box intuition model allowed more questions to an oracle for B (many-to-one reductions) (Turing reductions)
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  • Planar-3-COL INSTANCE: planar graph G QUESTION: can the vertices of G be assigned colors red,green,blue so that no two neighboring vertices have the same color?
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  • 3-COL Planar-3-COL
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  • 4-COL INSTANCE: graph G QUESTION: can the vertices of G be assigned one of 4 colors so that no two neighboring vertices have the same color?
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  • 3-COL 4-COL G G
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  • planar 4-COL INSTANCE: planar graph G QUESTION: can the vertices of G be assigned one of 4 colors so that no two neighboring vertices have the same color?
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  • planar 3-COL planar 4-COL ?
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  • planar 4-COL is very easy: the answer is always yes. (4-color theorem, Appel, Haken)
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  • Integer linear-programming INSTANCE: variables x 1,...,x n collection of linear inequalities over the x i with integer coefficients QUESTION: does there exist an assignment of integers to the x i such that all the linear inequalities are satisfied?
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  • Integer linear-programming INSTANCE: variables x 1,...,x n collection of linear inequalities over the x i with integer coefficients QUESTION: does there exist an assignment of integers to the x i such that all the linear inequalities are satisfied? x 1 1 x 2 16 x 1 x 3 16 x 2 x 4 16 x 3 x 3 +x 4 +x 1 10000
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  • Integer linear-programming we will show that ILP is NP-hard by showing 3-SAT ILP y 1 y 2 y 3 x 1 + (1-x 2 ) + x 3 1 0 x 1 1.... 0 x n 1 true = 1, false =0
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  • Integer linear-programming Is integer linear programming NP-complete ? I.e., is ILP in NP ? Witness of solvability = solution, but a priori we do not know that the solution is polynomially bounded. ILP NP, but the proof is far from trivial.
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  • Min-Cut problem cut S V number of edges crossing the cut | { {u,v} ; u S, v V-S } | INPUT: graph G OUTPUT: cut S with the minimum number of crossing edges
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  • Min-Cut problem in P for each s,t pair run max-flow algorithm
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  • Max-Cut problem cut S V number of edges crossing the cut | { {u,v} ; u S, v V-S } | INPUT: graph G OUTPUT: cut S with the maximum number of crossing edges
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  • Max-Cut problem INSTANCE: graph G, integer K QUESTION: does G have a cut with K crossing edges?
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  • Max-Cut problem INSTANCE: graph G, integer K QUESTION: does G have a cut with K crossing edges? NAE-3-SAT Max-Cut NAE-3-SAT INSTANCE: 3-CNF formula QUESTION: does there exist an assignment such that every claues have 1 false and 1 true ?
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  • NAE-3-SAT Max-Cut INSTANCE: 3-CNF formula QUESTION: does there exist an assignment such that every claues have 1 false and 1 true ? x 1 x 2 x 3 1 vertex for each literal x1x1 x2x2 x3x3 x2x2 2m parallel edges
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  • 3-SAT NAE-3-SAT y 1 y 2 y 3 y 1 y 2 z i, z i y 3 b 1.C satisfiable can find 3-NAE assignment for C 2.C has 3-NAE assignment C satisfiable
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  • Is NP co-NP = P ? Factoring INSTANCE: pair of integers n,k QUESTION: does n have a factor x {2,...k} ? Factoring decision version INPUT: integer n OUTPUT: factorization of n, i.e., n=p 1 1... p k k