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34.NP Completeness

34.NP Completeness. Hsu, Lih-Hsing. 為何學 NP-Complete. 50 年前的電腦 ( 一間房間大 ) 可解的問題的極限與現在電腦 ( 個人電腦 ) 可解的極限理論上是相同的 , 除非電腦的製程有重大突破 ( 即不是用目前半導體技術 ) 。 請問 C ， C++ ， java, Which one is best? 學習的態度是如果你發現你的問題是屬於 NP-C 則大部份的努力的方向是要放在找 Approximate algorithm 而不是 fast algorithm. P and NP.

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34.NP Completeness

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  1. 34.NP Completeness Hsu, Lih-Hsing

  2. 為何學 NP-Complete • 50 年前的電腦(一間房間大)可解的問題的極限與現在電腦(個人電腦)可解的極限理論上是相同的, 除非電腦的製程有重大突破(即不是用目前半導體技術)。 • 請問 C ，C++，java, Which one is best? • 學習的態度是如果你發現你的問題是屬於 NP-C則大部份的努力的方向是要放在找 Approximate algorithm 而不是 fast algorithm.

  3. P and NP Non-formal description • P: solvable polynomial time • NP: • nondeterministic polynomial time • Verifiable in polynomial time by deterministic Turing machine.

  4. NP-complete • NP-Complete: No polynomial-time algorithm has yet been discovered for an NP-computer problem, nor has anyone yet been able to prove that no polynomial-time algorithm can exist for any one of them.

  5. 千禧年大獎問題 • 2000年5月 • 7大問題懸賞求解100萬美元。由美國克萊數學院提供 • P 與 NP

  6. Given a description of a program and a finite input, decide whether the program Finishes running or will run forever.

  7. The subject of this chapter, however, is an interesting class of problems, called the “ NP-complete” problems, whose status is unknown. No polynomial-time algorithm has yet been discovered for an NP-computer problem, nor has anyone yet been able to prove that no polynomial-time algorithm can exist for any one of them. This so-called P ≠ NP question has been one of the deepest, most perplexing open research problems in theoretical computer science since it was first posed in 1971.

  8. The difference between these problems • Shortest vs. longest simple paths: • Euler tour vs. hamiltonian cycle: • 2-CNF satisfiability vs. 3 CNF satisfiablility • NP-completeness and the classes P and NP • Overview of showing problems to be NP-complete • Decision problems vs. optimization problems

  9. Reductions Suppose that there is a different decision problem, say B, that we already know how to solve in polynomial time. Finally, suppose that we have a procedure that transforms any instance  of A into some instance  of B with the following characteristics: 1.The transformation takes polynomial time. 2.The answer are the same. That is, the answer for  is “yes” if and only if the answer for  is also “yes.”

  10. yes yes   Polynomial-time reduction algorithm Polynomial-time Algorithm to decide B no no Polynomial-time algorithm to decide A We can call such a procedure a polynomial-time reduction algorithm and, it provides us a way to solve problem A in polynomial time: 1.Given an instance  of problem A, use a polynomial-time reduction algorithm to transform it to an instance  of problem B. 2.Run the polynomial-time decision algorithm for B on the instance . 3.Use the answer for  as the answer for .

  11. A First NP-complete problem • Because the technique of reduction relies on having a problem already known to be NP-complete in order to prove a different problem NP-complete, we need a “first” NPC problem. • Circuit-satisfiability problem

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