lecture 29

Oct 20, 2008Download as ppt, pdf

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The document discusses reductions between problems to prove NP-completeness. It first reviews P, NP, reductions, NP-hard and NP-complete problems. It then walks through reducing directed Hamiltonian cycle to undirected Hamiltonian cycle to traveling salesman problem (TSP) to prove that TSP is NP-complete in 3 steps or less.

CS 332: Algorithms NP Completeness Continued: Reductions

Homework 7 Check the web later today… Change in grading policy: drop lowest hw

Review: P And NP Summary P = set of problems that can be solved in polynomial time NP = set of problems for which a solution can be verified in polynomial time P  NP Open question: Does P = NP ?

Review: Reduction A problem P can be reduced to another problem Q if any instance of P can be rephrased to an instance of Q, the solution to which provides a solution to the instance of P This rephrasing is called a transformation Intuitively: If P reduces in polynomial time to Q, P is “no harder to solve” than Q

Review: NP-Hard and NP-Complete If P is polynomial-time reducible to Q, we denote this P  p Q Definition of NP-Hard and NP-Complete: If all problems R  NP are reducible to P, then P is NP-Hard We say P is NP-Complete if P is NP-Hard and P  NP If P  p Q and P is NP-Complete, Q is also NP- Complete

Review: Proving NP-Completeness What steps do we have to take to prove a problem Q is NP-Complete? Pick a known NP-Complete problem P Reduce P to Q Describe a transformation that maps instances of P to instances of Q, s.t. “yes” for Q = “yes” for P Prove the transformation works Prove it runs in polynomial time Oh yeah, prove Q  NP ( What if you can’t? )

Directed Hamiltonian Cycle  Undirected Hamiltonian Cycle What was the hamiltonian cycle problem again? For my next trick, I will reduce the directed hamiltonian cycle problem to the undirected hamiltonian cycle problem before your eyes Which variant am I proving NP-Complete? Draw a directed example on the board What transformation do I need to effect?

Transformation: Directed  Undirected Ham. Cycle Transform graph G = (V, E) into G’ = (V’, E’): Every vertex v in V transforms into 3 vertices v 1 , v 2 , v 3 in V’ with edges ( v 1 , v 2 ) and ( v 2 , v 3 ) in E’ Every directed edge ( v , w ) in E transforms into the undirected edge ( v 3 , w 1 ) in E’ (draw it) Can this be implemented in polynomial time? Argue that a directed hamiltonian cycle in G implies an undirected hamiltonian cycle in G’ Argue that an undirected hamiltonian cycle in G’ implies a directed hamiltonian cycle in G

Undirected Hamiltonian Cycle Thus we can reduce the directed problem to the undirected problem What’s left to prove the undirected hamiltonian cycle problem NP-Complete? Argue that the problem is in NP

Hamiltonian Cycle  TSP The well-known traveling salesman problem : Optimization variant: a salesman must travel to n cities, visiting each city exactly once and finishing where he begins. How to minimize travel time? Model as complete graph with cost c( i,j ) to go from city i to city j How would we turn this into a decision problem? A: ask if  a TSP with cost < k

Hamiltonian Cycle  TSP The steps to prove TSP is NP-Complete: Prove that TSP  NP ( Argue this ) Reduce the undirected hamiltonian cycle problem to the TSP So if we had a TSP-solver, we could use it to solve the hamilitonian cycle problem in polynomial time How can we transform an instance of the hamiltonian cycle problem to an instance of the TSP? Can we do this in polynomial time?

The TSP Random asides: TSPs (and variants) have enormous practical importance E.g., for shipping and freighting companies Lots of research into good approximation algorithms Recently made famous as a DNA computing problem

The document discusses NP-completeness and algorithms. It introduces dynamic programming and greedy algorithms. It then discusses the activity selection problem and shows it can be solved greedily by choosing the activity with the earliest finish time at each step. Finally, it defines the classes P and NP, introduces the concept of reductions to show problems are NP-complete, and states that if any NP-complete problem can be solved in polynomial time, then P=NP.

lecture 28sajinsc

This document discusses NP-completeness and algorithms. It begins by reviewing tractable versus intractable problems, and the complexity classes P and NP. P contains problems solvable in polynomial time, while NP includes problems verifiable in polynomial time. The document then discusses NP-complete problems, which are the hardest problems in NP. It notes that reducing one problem to another shows the first problem is no harder than the second. The document concludes by outlining the steps to prove a problem is NP-complete: choosing a known NP-complete problem, reducing it to the target problem in polynomial time, and showing the target is in NP.

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27 NP CompletnessAndres Mendez-Vazquez

lecture 30sajinsc

The document reviews concepts related to NP-completeness, including reductions between problems. It provides examples of reducing the directed Hamiltonian cycle problem to the undirected version. It also reduces 3-SAT to the clique problem by transforming a Boolean formula to a graph, then further reduces clique to vertex cover. Hundreds of problems have been shown to be NP-complete through relatively simple reductions like these that leverage previous results.

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np completeGayathri Gaayu

The document discusses three classes of decision problems: 1) P problems that can be solved quickly in polynomial time. 2) NP problems where a "YES" answer has a proof checkable in polynomial time. 3) co-NP problems where a "NO" answer has a proof checkable in polynomial time. It then defines NP-Complete problems as the hardest problems in NP, and explains that 3SAT is a famous NP-Complete problem involving finding a variable assignment that satisfies a Boolean formula of clauses with 3 variables each. The document provides methods for proving other problems like Clique and Independent Set are also NP-Complete by reducing 3SAT to them in polynomial time.

P versus NPFarid El Hajj

This document discusses the P versus NP problem in complexity theory. The P class contains problems that can be solved quickly by algorithms, while NP contains problems that can be verified quickly given a proposed solution. It is unknown whether NP problems can also be solved quickly (meaning P=NP), or if finding solutions to NP problems requires exponential time (meaning P≠NP). Solving this problem could have significant impacts on fields like cryptography, optimization, and theorem proving. However, despite extensive research, the P versus NP problem remains unsolved.

Np completenessRajendran

1) NP-Completeness refers to problems that are in NP (can be verified in polynomial time) and are as hard as any problem in NP. 2) The first problem proven to be NP-Complete was the Circuit Satisfiability problem, which asks whether there exists an input assignment that makes a Boolean circuit output 1. 3) To prove a problem P is NP-Complete, it must be shown that P is in NP and that any problem in NP can be reduced to P in polynomial time. This establishes P as at least as hard as any problem in NP.

Np completeMd. Shafiuzzaman Hira

This document discusses the complexity classes P, NP, and NP-Complete. P is the class of problems solvable in polynomial time. NP is the class of problems verifiable in polynomial time. NP-Complete problems are the hardest problems in NP, such that any NP problem can be reduced to an NP-Complete problem in polynomial time. Some examples of problems in each class are given, such as fractional knapsack in P and NP, and SAT and the traveling salesman problem in NP-Complete. It is still an open question whether P equals NP.

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The document summarizes the concepts of P vs NP complexity classes. It states that P problems can be solved in polynomial time, like searching an array, while NP problems are solved in non-deterministic polynomial time, like the knapsack problem. It then defines different types of algorithms and complexity classes. The key classes discussed are P, NP, NP-Complete, and NP-Hard. It provides examples like sorting being in P, while the Hamiltonian problem is NP-Complete. A graphical representation is also included to illustrate the relationships between the complexity classes.

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NP-completeness refers to problems that are both in the complexity class NP and NP-hard. These problems are considered intractable because they have no known efficient polynomial-time solutions. The document lists several classic NP-complete problems, including Boolean satisfiability, the traveling salesman problem, and Hamiltonian path problem. It also defines polynomial-time tractability and explains that NP-complete problems are harder than problems that are provably intractable.

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This document discusses P, NP and NP-complete problems. It begins by introducing tractable and intractable problems, and defines problems that can be solved in polynomial time as tractable, while problems that cannot are intractable. It then discusses the classes P and NP, with P containing problems that can be solved deterministically in polynomial time, and NP containing problems that can be solved non-deterministically in polynomial time. The document concludes by defining NP-complete problems as those in NP that are as hard as any other problem in the class, in that any NP problem can be reduced to an NP-complete problem in polynomial time.

Np completenessMuhammad Saim

The document introduces the concepts of P, NP, and NP-completeness. It states that P problems can be solved in polynomial time, while NP problems can be verified in polynomial time using a non-deterministic algorithm. NP-complete problems are the hardest problems in NP and if any NP-complete problem could be solved in polynomial time, then all NP problems could be solved that way. As an example, it describes how the Traveling Salesman Problem is NP-complete as the number of possible routes grows exponentially with the number of cities.

Tsp is NP-CompleteEmre Can Kucukoglu

This document discusses how the Traveling Salesman Problem (TSP) is NP-Complete. It first shows that TSP is in NP by describing a nondeterministic polynomial time algorithm to solve it. It then reduces the known NP-Complete Hamiltonian Cycle problem to TSP by constructing an equivalent instance of TSP from any Hamiltonian Cycle problem instance in polynomial time, showing that Hamiltonian Cycle is polynomial time reducible to TSP. Therefore, since any problem in NP can be reduced to Hamiltonian Cycle and Hamiltonian Cycle can be reduced to TSP, any problem in NP can be reduced to TSP, proving that TSP is NP-Complete.

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This document discusses NP-complete problems and their properties. Some key points: - NP-complete problems have an exponential upper bound on runtime but only a polynomial lower bound, making them appear intractable. However, their intractability cannot be proven. - NP-complete problems are reducible to each other in polynomial time. Solving one would solve all NP-complete problems. - NP refers to problems that can be verified in polynomial time. P refers to problems that can be solved in polynomial time. - A problem is NP-complete if it is in NP and all other NP problems can be reduced to it in polynomial time. Proving a problem is NP-complete involves showing

A comprehensive view on P vs NPAbhay Pai

This document provides a comprehensive overview of the P vs NP problem. It discusses the problem statement, definitions of complexity classes P and NP, methods that have been used to attempt to solve P vs NP like diagonalization and relativization, computational circuits, approximation algorithms, quantum computing, and geometric complexity theory. It also mentions surveys by Sipser and Fortnow on P vs NP, as well as statistics on opinions of experts about whether P equals NP or not. The document concludes that P vs NP has significantly impacted computer science and other fields and that more work is still needed to solve this important open problem.

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This document introduces the concept of NP-completeness. It discusses that while some problems like shortest paths, minimum spanning trees, and bipartite matching have efficient polynomial time algorithms, other problems like satisfiability (SAT), the travelling salesman problem (TSP), integer linear programming (ILP), and set cover have only exponential time algorithms. It defines the class NP as problems that can be solved by a non-deterministic Turing machine in polynomial time. It states that if any NP-complete problem could be solved in polynomial time, then P would equal NP. Problems are NP-hard if all problems in NP can be reduced to them in polynomial time, and NP-complete if they are both in NP and NP-

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lecture 29

1. CS 332: Algorithms NP Completeness Continued: Reductions

2. Homework 7 Check the web later today… Change in grading policy: drop lowest hw

3. Review: P And NP Summary P = set of problems that can be solved in polynomial time NP = set of problems for which a solution can be verified in polynomial time P  NP Open question: Does P = NP ?

4. Review: Reduction A problem P can be reduced to another problem Q if any instance of P can be rephrased to an instance of Q, the solution to which provides a solution to the instance of P This rephrasing is called a transformation Intuitively: If P reduces in polynomial time to Q, P is “no harder to solve” than Q

5. Review: NP-Hard and NP-Complete If P is polynomial-time reducible to Q, we denote this P  p Q Definition of NP-Hard and NP-Complete: If all problems R  NP are reducible to P, then P is NP-Hard We say P is NP-Complete if P is NP-Hard and P  NP If P  p Q and P is NP-Complete, Q is also NP- Complete

6. Review: Proving NP-Completeness What steps do we have to take to prove a problem Q is NP-Complete? Pick a known NP-Complete problem P Reduce P to Q Describe a transformation that maps instances of P to instances of Q, s.t. “yes” for Q = “yes” for P Prove the transformation works Prove it runs in polynomial time Oh yeah, prove Q  NP ( What if you can’t? )

7. Directed Hamiltonian Cycle  Undirected Hamiltonian Cycle What was the hamiltonian cycle problem again? For my next trick, I will reduce the directed hamiltonian cycle problem to the undirected hamiltonian cycle problem before your eyes Which variant am I proving NP-Complete? Draw a directed example on the board What transformation do I need to effect?

8. Transformation: Directed  Undirected Ham. Cycle Transform graph G = (V, E) into G’ = (V’, E’): Every vertex v in V transforms into 3 vertices v 1 , v 2 , v 3 in V’ with edges ( v 1 , v 2 ) and ( v 2 , v 3 ) in E’ Every directed edge ( v , w ) in E transforms into the undirected edge ( v 3 , w 1 ) in E’ (draw it) Can this be implemented in polynomial time? Argue that a directed hamiltonian cycle in G implies an undirected hamiltonian cycle in G’ Argue that an undirected hamiltonian cycle in G’ implies a directed hamiltonian cycle in G

9. Undirected Hamiltonian Cycle Thus we can reduce the directed problem to the undirected problem What’s left to prove the undirected hamiltonian cycle problem NP-Complete? Argue that the problem is in NP

10. Hamiltonian Cycle  TSP The well-known traveling salesman problem : Optimization variant: a salesman must travel to n cities, visiting each city exactly once and finishing where he begins. How to minimize travel time? Model as complete graph with cost c( i,j ) to go from city i to city j How would we turn this into a decision problem? A: ask if  a TSP with cost < k

11. Hamiltonian Cycle  TSP The steps to prove TSP is NP-Complete: Prove that TSP  NP ( Argue this ) Reduce the undirected hamiltonian cycle problem to the TSP So if we had a TSP-solver, we could use it to solve the hamilitonian cycle problem in polynomial time How can we transform an instance of the hamiltonian cycle problem to an instance of the TSP? Can we do this in polynomial time?

12. The TSP Random asides: TSPs (and variants) have enormous practical importance E.g., for shipping and freighting companies Lots of research into good approximation algorithms Recently made famous as a DNA computing problem

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