L13-duality - LP Duality Le c ture 1 3 :Fe b 2 8 1 Min-Max...

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1 LP Duality Le c ture  13: Fe b  28
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2 Min-Max Theorems Min-Max Theorems In b ipa rtite  g raph,    Maximum matching = Minimum Vertex Cover In e ve ry g ra ph,   Maximum Flow = Minimum Cut Bo th the s e  re latio ns  c an be  de rive d fro m  the  c o m b ina to rial a lg o rithm s . We ’ve  als o  s e e n ho w to  s o lve  the s e  pro ble m s  b y line a r pro g ra m m ing . C a n we  a ls o  o b ta in the s e  m in-m ax the o re m s  fro m  line ar pro g ram m ing ? Ye s ,  LP-duality theorem .
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3 Example Example Is  o ptim a l s o lutio n <= 30? Ye s , c o ns ide r (2,1,3)
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4 NP and co-NP? NP and co-NP? Uppe r bo und is  e as y to  “pro ve ”,  we  jus t ne e d to  g ive  a  s o lutio n. Wha t a b o ut lo we r bo unds ? This  s ho ws  tha t the  pro b le m  is  in NP.
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Example Example Is  o ptim a l s o lutio n >= 5? Ye s , b e c aus e  x3 >= 1. Is  o ptim a l s o lutio n >= 6? Ye s , be c aus e  5x1 + x2 >= 6. Is  o ptim a l s o lutio n >= 16?
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L13-duality - LP Duality Le c ture 1 3 :Fe b 2 8 1 Min-Max...

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