CS300-12_NP_Complete_Problems - NP - Complete Problems Sung...

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NP - Complete Problems Sung Yong Shin TC Dept., KAIST
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Introduction Travelling Salesman Problem B A C D E F G H I
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Serious damage on your position within the company !!! “I can’t find an efficient algorithm, I guess I’m just too dumb.”
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“I can’t find an efficient algorithm, because no such algorithm is possible!” Unfortunately,proving intractability can be just as hard as finding efficient algorithms !!! No hope !!! P NP
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“I can’t find an efficient algorithm, but neither can all these famous people”
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“Anyway, we have to solve this problem. Can we satisfy with a good solution ?
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10 20 30 40 50 60 .00001 .00002 .00003 .00004 .00005 .00006 second second second second second second .0001 .0004 .0009 .0016 .0025 .0036 second second second second second second .001 .008 .027 .064 .125 .216 second second second second second second .1 3.2 24.3 1.7 5.2 13.0 second second second minutes minutes minutes .001 1.0 17.9 12.7 35.7 366 second second minutes days years centuries .059 58 6.5 3855 2*10 8 1.3 *10 13 second minutes years centuries centuries centuries n n 2 n 3 n 5 2 n 3 n Size n Time complexity function Comparison of several polynomial and exponential time complexity functions
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n n 2 n 3 n 5 2 n 3 n N 1 N 2 N 3 N 4 N 5 N 6 100N 1 10N 2 4.64N 3 2.5N 4 N 5 +6.64 1000N 1 31.6N 2 10N 3 3.98N 4 N 6 +4.19 N 5 +9.97 N 6 +6.29 Time complexity function With present computer With computer 100 times faster With computer 1000 times faster Size of Largest Problem Instance Solvable in 1 Hour
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Intractable problems procedure Algorithm note r.e. sets decidable non r.e. sets undecidable not always halts always halts yes yes no recursive sets recursively enumerable sets
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Definition : A problem is said to be intractable if it is so hard that no polynomial time algorithms can possibly solve it. Certain decidable problems may be intractable !!! What are polynomial time algorithms ? Why “possibly” ? In what ? In which computer ?
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problem Given I and P , find S satisfying P. A generic instance is specified in terms of various components (parameters) such as sets,graphs,functions,numbers , …. A question is asked in terms of the generic instance. Decision problems Optimization problems “yes” or “no” answers Generic instance Question problem
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Traveling Salesman Instance: A finite set C = {c 1 , c 2 ,…., c m } of cities , a distance d(c i , c j ) Z + for each pair of cities c i , c j C, and a bound B Z + (where Z + denote the positive integers ) Question: Is there a “tour” of all cities in C having total length no more than B. < c π (1) , c π (2) , …. , c π (m) > such that . ) , ( ) , ( ) 1 ( ) ( 1 1 ) 1 ( ) ( B c c d c c d m m i i i + - = + π “yes “ or “no”
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Algorithm a TM program that always halts An algorithm is a step - by - step porcedure( consisting of a finite sequence of instructions) for solving a problem. Polynomial time algorithms exponential time algorithms no yes
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Time Complexity T(n) T(n) time requirements of an algorithm for solving a problem of size n Need two underlying assumptions: (i) The model of computation TM (ii) a reasonable encoding scheme ?
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This note was uploaded on 02/04/2010 for the course COMPUTER S cs300 taught by Professor Unkown during the Spring '08 term at Korea Advanced Institute of Science and Technology.

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CS300-12_NP_Complete_Problems - NP - Complete Problems Sung...

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