NP-Complete problems NP-Complete problems - Ace
NP-Complete problems NP-Complete problems - Ace
NP-Complete problems NP-Complete problems - Ace
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✬✩Alternative definition of <strong>NP</strong>-completenessLemma 34.8: If L is a language such that L’ ≤ p L for some L’∈ <strong>NP</strong>C, then L is <strong>NP</strong>-hard. Moreover, if L ∈ <strong>NP</strong>, then L ∈ <strong>NP</strong>C.Proof: Since L’ is <strong>NP</strong>-complete, for all L” ∈ <strong>NP</strong>, we have L”≤ p L’. By supposition, L’ ≤ p L, and thus by transitivity, wehave L” ≤ p L, which shows that L is <strong>NP</strong>-hard. If L ∈ <strong>NP</strong>, thenwe also have L ∈ <strong>NP</strong>C.Transitivity: If L 1 ≤ p L 2 and L 3 ≤ p L 3 , then L 1 ≤ p L 3 (Exercise34.3-2).CS404/504✫Design and Analysis of Algorithms: Lecture 26Computer Science13✪
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