The P versus NP Problem. Ker-I Ko. Stony Brook, New York

Transcription

1 The P versus NP Problem Ker-I Ko Stony Brook, New York

2 ? P = NP One of the seven Millenium Problems The youngest one A folklore question? Has hundreds of equivalent forms

3 Informal Definitions P : Computational problems whose solutions are easy to find ( feasibly solvable problems). NP: Computational problems whose solutions are easy to verify but not necessarily easy to find. NP-Complete: Hardest problems in NP. Easy means it can be done in a polynomial number of steps (i.e., in O(n c ) steps on inputs of size n).

4 An example: Sudoku Fill in each cell with a digit from 1 to 9 so that each row, each column, and each region contains exactly one instance of each digit

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6 Sudoku is a special case of Graph Coloring: Given a graph G = (V, E), and an integer k > 0, determine whether there is a coloring c of vertices using k colors such that no two adjacent vertices have the same color. Complexity measure: based on the number of vertices n = V. Graph Coloring is in NP: For any given coloring c (a potential solution), just verify that, for each edge {u, v}, c(u) c(v). It only takes O(n 2 ) steps.

7 Is Graph Coloring in P? There are k n potential solutions. An exhaustive search would take 100,000 centuries to solve an instance of size n = 50 and k = 3, assuming that one can verify each coloring in 10 second. If we can solve Graph Coloring easily, then we can solve Sudoku easily. Graph Coloring is NP-complete. (Actually, it remains NP-complete even if k is a fixed constant 3.) So, Graph Coloring P P = NP.

8 An example from Number Theory Composite Number: Given an integer n in its binary expansion, determine whether n is a composite number. Complexity measure: log n number of binary digits in n. Composite Number is in NP: For any potential factor k, we can easily verify whether k divides n. Is Composite Number in P? YES, [Agarwal-Kayal-Saxena, 2002].

9 An informal example in Mathematical Logic: For a fixed proof system S, let T S be the set of all theorems in S that have a short proof. T S is in NP. Is T S in P? If P = NP then there is an efficient automatic theorem prover that can prove all theorems with short proofs.

10 NP-complete problems are ubiquitous (over 2000 are known): From Mathematical Programming: Integer Programming: Given a set of linear inequalities, find an integer solution to it. Linear Programming (which allows real-valued solutions) is in P [Khachian, 1979; Karmarkar, 1984].

11 From Graph Theory: Hamiltonian Circuit: Given a graph G, find a tour of the graph that visits each vertex exactly once. A famous variation is Traveling Salesman: Given an edge-weighted complete graph G, find a Hamiltonian tour with the minimum total edge weight. The problem Euler Circuit (also known as the Königsberg Bridge problem) that asks for a tour of a graph that passes through each edge exactly once is in P.

12 A special case of Hamiltonian Circuit: Knight s Tour: Move a Knight through an n n chessboard so that it visits each square exactly once

13 From Mathematical Logic: Satisfiability: Given a Boolean formula, find a Boolean assignment to the variables to satisfy the formula. Input: Φ = (x 1 + x 2 + x 3 )(x 1 + x 3 + x 4 )(x 2 + x 3 + x 4 ) Question: Find τ : {x 1,..., x n } {True, False} that makes Φ True. Use truth table?? (size = 2 n is too big) This is the first problem that is proved to be NP-complete [Cook, 1970].

14 From Number Theory: Integer Factoring: Given an integer n > 0 (written in binary notation), find its prime factors. Integer Factoring is not known to be NP-complete, nor known to be in P. Composite Number is known to be in P. The famous RSA public-key cryptosystem is based on Integer Factoring.

15 Games and Puzzles (Generalized Geography) Combinatorics (Set Partition) Network Design (Steiner Tree) Sequencing and Scheduling (Longest Common Subsequence) Automata Theory (Finite State Automata Inequivalence) Combinatorial Optimization (Knapsack) Numerical Analysis (Maximization)

16 Significance of P = NP All problems in NP are feasibly solvable. Software breakthrough. Chess programs beating human players. Automated theorem provers. Internet security breakdown.

17 Significance of P NP Breakthrough in mathematical logic; new proof techniques A more accurate classification of computational complexity Breakthrough in cryptography

18 Current Research about NP-completeness Algorithmic Approach To develop new algorithmic techniques on specific NP-complete problems Heuristic algorithms (greedy algorithms, dynamic programming) Approximation algorithms (linear programming, relaxation, restriction) Subproblems (planar graphs, fixed parameters) Randomized algorithms (pseudorandom generator, derandomization)

19 Computational Models To study different models and develop new proof techniques Circuit Complexity (lower bound techniques) Communication Complexity (information-theoretic techniques) Decision Tree Complexity (algebraic techniques) Probabilistic Complexity (probability theory, algebraic coding theory) Quantum Complexity (quantum theory) Kolmogorov Complexity (mathematical logic, probability theory)

20 Structural Analysis To understand the mathematical properties of hard problems Diagonalization (set theory) Completeness and Reducibility (recursion theory) Complexity Classes (space complexity) Isomorphism Conjecture Relativized Computation Average-case Complexity (probability theory) Instance Complexity