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, 3-COL

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)

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?

3-COL  Planar-3-COL

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?

3-COL  4-COL G  G

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?

planar 3-COL  planar 4-COL ?

planar 4-COL is very easy: the answer is always yes. (4-color theorem, Appel, Haken)

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?

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

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   x n  1 true = 1, false =0

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.

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

Min-Cut problem in P for each s,t pair run max-flow algorithm

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

Max-Cut problem INSTANCE: graph G, integer K QUESTION: does G have a cut with  K crossing edges?

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 ?

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 x2x2 x3x3 x2x2 2m parallel edges

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

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