Algorithm runs in time polynomial in n X x c x element of cost paid only when x

Algorithm runs in time polynomial in n x x c x

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Algorithm runs in time polynomial in n . X x c x element of cost paid only when x is covered for the first time ) ( 1 1 2 1 i i x S S S S c assume x is covered for the first time by S i (spread cost evenly across all elements covered for first time by S i ) Number of elements covered for first time by S i ) ( 1 number harmonic th 1 d H i H d d i d
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Set Cover (proof continued) Proof : (continued) *   C S S x x c C   X x x C S S x x c c * *   C S S x x c Cost assigned to optimal cover: Each x is in >= 1 S in C* COVER - SET - GREEDY from cover be cover optimal an be * Let C C X x x c C 1 unit is charged at each stage of algorithm Theorem : GREEDY-SET-COVER is a polynomial-time (n)-approximation algorithm for }) |: (max{| ) ( F S S H n
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Set Cover (proof continued) Proof : (continued) ) ( S H c S x x How does this relate to harmonic numbers?? ) ( 1 number harmonic th 1 d H i H d d i d We’ll show that: }) : (max{ * ) ( * F S S H C S H C C S And then conclude that: which will finish the proof Theorem : GREEDY-SET-COVER is a polynomial-time (n)-approximation algorithm for }) |: (max{| ) ( F S S H n For any set S
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