Fast Exponential-Time Algorithms to solve NP-complete ... - Lita

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7/52Fast Exponential-Time AlgorithmsIterative CompressionCompression (Expansion)Two possible cases1. α(G i+1 ) = α(G i ) and B i+1 = B i .2. α(G i+1 ) = α(G i ) + 1, v i+1 ∈ B i+1 . ThusB i+1 = {v i+1 } ∪ A ′ ∪ B ′ , where A ′ ⊆ A i , B ′ ⊆ B i and A ′independent set.To find out whether α(G i+1 ) = α(G i ) + 1, compute for everymaximal independent set A ′ of G[A i − N[v i+1 ]] in time O(n 2.5 ) amaximum independent set of the bipartite graphG[(A ′ ∪ B i ) − N[v i+1 ]].
8/52Fast Exponential-Time AlgorithmsIterative CompressionSpeed up the CompressionLet |A i | = an and |B i | ≤ (1 − a)n. Depending on the value of a :1. Either check all at most 3 an/3 maximal independent sets A ′ ofG[A i ]2. Or check all independent sets A ′ ⊆ A i − N[v i+1 ] with |A ′ | ≤ |B i |Running times (up to polynomial factor) :1. (3 a/3 ) n2. ( )an(1−a)nBalancing with a ≈ 0.8 : MAXIMUM INDEPENDENT SET can besolved without Branching in time O(1.3196 n ).
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Fast Exponential-Time Algorithms to solve NP-complete ... - Lita

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