Note on Set Cover
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چکیده
Greedy: Θ(min(logd,q)): Θ(H(d)) (this also implies O(logn), n is number of elements). (Note: Feige [3] prove that there is no substantially improvement (for greedy algo).) Also O(q) and exists tight example (section 2.5); so, Θ(q) in general. Note that this bound is asymptotically tight (see Vazirani’s book [10] section 29.9) Slav́ık [9] also gives exact analysis lnn− lnlnn+Θ(1). LP Rounding: O(f) (see section 3.3) by Hochbaum [6]. Bar-Yehuda and Kehat [1] also improve to tighter bound ( k k−(k−1) k √ 1−α ). Special cases In dense case, where every element is in large portion of sets (i.e., fe ≥ α|S|), Karpinski and Zelikovsky [8] show O(log 1 1−α n) (see section 5.1). Bar-Yehuda and Kehat [1] also claim a tighter bound. Variants
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