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CSE 421 Algorithms Richard Anderson Lecture 27 NP Completeness

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Algorithms vs. Lower bounds Algorithmic Theory –What we can compute I can solve problem X with resources R –Proofs are almost always to give an algorithm that meets the resource bounds Lower bounds –How do we show that something can’t be done?

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Theory of NP Completeness

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The Universe NP-Complete NP P

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Polynomial Time P: Class of problems that can be solved in polynomial time –Corresponds with problems that can be solved efficiently in practice –Right class to work with “theoretically”

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What is NP? Problems solvable in non-deterministic polynomial time... Problems where “yes” instances have polynomial time checkable certificates

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Decision Problems Theory developed in terms of yes/no problems –Independent set Given a graph G and an integer K, does G have an independent set of size at least K –Vertex cover Given a graph G and an integer K, does the graph have a vertex cover of size at most K.

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Certificate examples Independent set of size K –The Independent Set Satifisfiable formula –Truth assignment to the variables Hamiltonian Circuit Problem –A cycle including all of the vertices K-coloring a graph –Assignment of colors to the vertices

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Polynomial time reductions Y is Polynomial Time Reducible to X –Solve problem Y with a polynomial number of computation steps and a polynomial number of calls to a black box that solves X –Notations: Y < P X

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Lemma Suppose Y < P X. If X can be solved in polynomial time, then Y can be solved in polynomial time.

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Lemma Suppose Y < P X. If Y cannot be solved in polynomial time, then X cannot be solved in polynomial time.

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NP-Completeness A problem X is NP-complete if –X is in NP –For every Y in NP, Y < P X X is a “hardest” problem in NP If X is NP-Complete, Z is in NP and X < P Z –Then Z is NP-Complete

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Cook’s Theorem The Circuit Satisfiability Problem is NP- Complete

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Garey and Johnson

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History Jack Edmonds –Identified NP Steve Cook –Cook’s Theorem – NP-Completeness Dick Karp –Identified “standard” collection of NP-Complete Problems Leonid Levin –Independent discovery of NP-Completeness in USSR

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Populating the NP-Completeness Universe Circuit Sat < P 3-SAT 3-SAT < P Independent Set 3-SAT < P Vertex Cover Independent Set < P Clique 3-SAT < P Hamiltonian Circuit Hamiltonian Circuit < P Traveling Salesman 3-SAT < P Integer Linear Programming 3-SAT < P Graph Coloring 3-SAT < P Subset Sum Subset Sum < P Scheduling with Release times and deadlines

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Cook’s Theorem The Circuit Satisfiability Problem is NP- Complete Circuit Satisfiability –Given a boolean circuit, determine if there is an assignment of boolean values to the input to make the output true

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Circuit SAT AND OR AND OR ANDNOTOR NOT x1x1 x2x2 x3x3 x4x4 x5x5 AND NOT AND OR NOT AND OR AND Find a satisfying assignment

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Proof of Cook’s Theorem Reduce an arbitrary problem Y in NP to X Let A be a non-deterministic polynomial time algorithm for Y Convert A to a circuit, so that Y is a Yes instance iff and only if the circuit is satisfiable

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Satisfiability Given a boolean formula, does there exist a truth assignment to the variables to make the expression true

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Definitions Boolean variable: x 1, …, x n Term: x i or its negation !x i Clause: disjunction of terms –t 1 or t 2 or … t j Problem: –Given a collection of clauses C 1,..., C k, does there exist a truth assignment that makes all the clauses true –(x 1 or !x 2 ), (!x 1 or !x 3 ), (x 2 or !x 3 )

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3-SAT Each clause has exactly 3 terms Variables x 1,..., x n Clauses C 1,..., C k –C j = (t j1 or t j2 or t j3 ) Fact: Every instance of SAT can be converted in polynomial time to an equivalent instance of 3-SAT

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Find a satisfying truth assignment (x || y || z) && (!x || !y || !z) && (!x || y) && (x || !y) && (y || !z) && (!y || z)

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Theorem: CircuitSat < P 3-SAT

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Theorem: 3-SAT < P IndSet

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Sample Problems Independent Set –Graph G = (V, E), a subset S of the vertices is independent if there are no edges between vertices in S 1 3 2 6 7 4 5

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Vertex Cover –Graph G = (V, E), a subset S of the vertices is a vertex cover if every edge in E has at least one endpoint in S 1 3 2 6 7 4 5

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IS < P VC Lemma: A set S is independent iff V-S is a vertex cover To reduce IS to VC, we show that we can determine if a graph has an independent set of size K by testing for a Vertex cover of size n - K

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IS < P VC 1 3 2 6 7 4 5 1 3 2 6 7 4 5 Find a maximum independent set S Show that V-S is a vertex cover

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Clique –Graph G = (V, E), a subset S of the vertices is a clique if there is an edge between every pair of vertices in S 1 3 2 6 45 7

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Complement of a Graph Defn: G’=(V,E’) is the complement of G=(V,E) if (u,v) is in E’ iff (u,v) is not in E 1 3 2 6 7 4 5 1 3 2 6 7 4 5 Construct the complement

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IS < P Clique Lemma: S is Independent in G iff S is a Clique in the complement of G To reduce IS to Clique, we compute the complement of the graph. The complement has a clique of size K iff the original graph has an independent set of size K

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Hamiltonian Circuit Problem Hamiltonian Circuit – a simple cycle including all the vertices of the graph

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Thm: Hamiltonian Circuit is NP Complete Reduction from 3-SAT

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Traveling Salesman Problem Given a complete graph with edge weights, determine the shortest tour that includes all of the vertices (visit each vertex exactly once, and get back to the starting point) 1 2 4 2 3 5 4 7 7 1 Find the minimum cost tour

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NP-Completeness Reductions If X is NP-Complete, Y is in NP, and X < P Y, then Y is NP-Complete

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Hamiltonian Circuit, Hamiltonian Path How do you show that Hamiltonian Path is NP-Complete?

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Local Modification Convert G to G’ Pick a vertex v –Replace v by v’ and v’’ –If (u,v) is an edge, include edges (u, v’), (u, v’’) G’ has a Hamiltonian Path from v’ to v’’ iff G has a Hamiltonian Circuit

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HamPath < P DirHamPath How do you show that Directed Hamiltonian Path is NP-Complete?

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Problem definition Given a graph G, does G have an independent set? Given a graph G, does G have an independent set of size 7? Given a graph G, and an integer K, does G have an independent set of size K?

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Graph Coloring NP-Complete –Graph K-coloring –Graph 3-coloring Polynomial –Graph 2-Coloring

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Number Problems Subset sum problem –Given natural numbers w 1,..., w n and a target number W, is there a subset that adds up to exactly W? Subset sum problem is NP-Complete Subset Sum problem can be solved in O(nW) time

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Subset sum problem The reduction to show Subset Sum is NP- complete involves numbers with n digits In that case, the O(nW) algorithm is an exponential time and space algorithm

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What is NP? Problems where ‘yes’ instances can be efficiently verified –Hamiltonian Circuit –3-Coloring –3-SAT Succinct certificate property

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What about ‘negative instances’ How do you show that a graph does not have a Hamiltonian Circuit How do you show that a formula is not satisfiable?

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What we don’t know P vs. NP NP-Complete NP P NP = P

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