P versus NP a Millennium Prize Problem

P versus NPa Millennium Prize Problem Chung-Chih Li School of Information Technology, ISU March 21, 2018 STV 346 Math Talk

7 Millennium Problems Clay Mathematics Institute • Yang–Mills and Mass Gap • Riemann Hypothesis • P vs NP Problem • Navier–Stokes Equation • Hodge Conjecture • Poincaré Conjecture(solved, Grigori Perelman/Richard Hamilton) • Birch and Swinnerton-Dyer Conjecture Math Talk

The hardness of a problem  whether an efficient algorithm exists • Binary Search, O(log n), supper fast • Sequential Search, O(n), fast • Sorting, O(n log n), good • Shortest Path, O(n2), OK • Minimum spanning tree, O(n2), OK • Maximum flow, not so bad • Euler Path, not so bad • Clique, not so good • Hamiltonian path, not good • Vertex cover, feel bad, • SAT, bad • ….. What is going on? Math Talk

Boolean formula: f(a,b,c) Satifibility (SAT): is a given Boolean formula satifiable? a a = T a = F b b b = T b = T b = F b = F c c c c c = T c = T c = F c = T c = F c = F c = F c = T T F F F F T F F  f(a,b,c) Math Talk

Boolean formula: f(a,b,c) Backtracking algorithm a a = T a = F Exp b b b = T b = T b = F b = F c c c c c = T c = T c = F c = T c = F c = F c = F c = T T F F F F T F F  f(a,b,c) Math Talk

P << Exp Math Talk

Nondeterministic algorithm Boolean formula: f(a,b,c) a a = T a = F b b b = T b = T b = F b = F N P c c c c c = T c = T c = F c = T c = F c = F c = F c = T T F F F F T F F If one of the result is T, then f is satisfiable. f(a,b,c) Math Talk

Nondeterministic algorithms are not randomized algorithms. Nondeterministic algorithms can’t be implemented using parallel computers. Nondeterministic algorithms can’t be implemented using quantum computers. a a = T a = F b b b = T b = T b = F b = F c c c c c = T c = T c = F c = T c = F c = F c = F c = T T F F F F T F F f(a,b,c) Math Talk

Is there a 3-clique? v1 v2 v5 v1v2v3 v4 v3 v2v3v4 v1v2v4 v1v3v4 v1v2v5 NP v1v3v5 v1v4v5 v3v4v5 v2v4v5 v2v3v5 Math Talk

Is there a 2-vertex-cover? Is there a 3-clique? v1 v1 v5 v2 v2 v2 v5 v2 p v4 v4 v3 v4 v4 v3  complement graph  this can be done in P Math Talk

Is there A  S, such that A = 2730 ? Is there a 2-vertex-cover? S = {1, 4, 16, 64, 256, 1040, 1089, 1108, 1280, 1285 } v1 v5 v2 v4 v3 A = {1, 16, 64, 256, 1108, 1285} How about  A = 1706 ? Math Talk

Is there a 2-vertex-cover? v1 e1 e2 v5 v2 e5 e3 e4 v4 v3 Math Talk

(x  y z)  (x   y z)  ( x  y  z) Is there a 9-vertex-cover? x z y z x y y z z x z z Math Talk

Cook Reduction, Polynomial Time Reduction A p B A is reduced to B B reduces A SAT p Vertex-Cover Clique p Vertex-Cover Clique p Vertex-Cover Vertex-Cover p Clique Math Talk

Problem A is called NP-complete if A is NP Every NP problem can be reduced to A Steven Cook (1971): SAT is an NP-Complete Problem. Math Talk

P versus NP a Millennium Prize Problem