L02-NP - NPcompleteness Lecture2:Jan11 1 P T h e c la s s o...

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1 NP-completeness Lecture 2: Jan 11
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2 P The  c las s  o f pro b le m s  that c a n be  s o lve d in po lyno m ia l tim e . e .g . g c d, s ho rte s t pa th, prim e , e tc . The re  are  m a ny pro b le m s  tha t we  do n’t kno w ho w to  s o lve  in po lyno m ial tim e . e .g . fac to ring , po lyno m ial ide ntitie s , g ra ph c o lo uring , e tc .
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3 NP-completeness
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4 NP-completeness
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5 NP-completeness
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6 Polynomial Time Reduction Ho w to  s ho w that a  pro b le m  R is  no t e a s ie r tha n a  pro b le m  Q ? Info rm a lly, if R c a n be  s o lve d e ffic ie ntly, we  c a n s o lve  Q  e ffic ie ntly. Fo rm ally, we  s a y Q  po lyno m ially re duc e s  to  R if: 1. G ive n a n ins ta nc e  q  o f pro b le m  Q 1. The re  is  a  po lyno m ial tim e  trans fo rm a tio n to  an ins tanc e  f(q ) o f R 1. q  is  a “ye s ” ins ta nc e  if and o nly if f(q ) is  a  “ye s ” ins tanc e   The n, if R is  po lyno m ia l tim e  s o lva ble , the n Q  is  po lyno m ia l tim e  s o lva ble . If Q  is  no t po lyno m ial tim e  s o lva b le , the n R is  no t po lyno m ial tim e  s o lvab le .
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First Example Clique:   a s ub s e t o f ve rtic e s  S  s o  that fo r e ve ry               two  ve rtic e s  u,v in S  a re  jo ine d by an e dg e . Ins ta nc e :     A g ra ph G =(V,E) a nd a po s itive  inte g e r k. Q ue s tio n:    Is  the re  a  c liq ue  o f s ize  k o r m o re  fo r G ? Independent set:   a  s ubs e t o f ve rtic e s  S  s o  tha t fo r e ve ry                                two  ve rtic e s  u,v in S  a re  jo ine d by an e dg e . Ins ta nc e :     A g raph G =(V,E) and a  po s itive  inte g e r k. Q ue s tio n:    Is  the re  a n inde pe nde nt o f s ize  k o r m o re  fo r G ?
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L02-NP - NPcompleteness Lecture2:Jan11 1 P T h e c la s s o...

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