{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

CS300-12_NP_Complete_Problems

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

This preview shows pages 1–15. Sign up to view the full content.

NP - Complete Problems Sung Yong Shin TC Dept., KAIST

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Introduction Travelling Salesman Problem B A C D E F G H I
Serious damage on your position within the company !!! “I can’t find an efficient algorithm, I guess I’m just too dumb.”

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
“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
“I can’t find an efficient algorithm, but neither can all these famous people”

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
“Anyway, we have to solve this problem. Can we satisfy with a good solution ?
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 ?
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 π π π “yes “ or “no”
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 ?
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 91

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

This preview shows document pages 1 - 15. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online