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

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21/52Fast Exponential-Time AlgorithmsInclusion ExclusionInclusion Exclusion FormulaFor a positive integer k, let c k = c k (S) denote the number of(possibly overlapping) k-covers, that is the number of ways tochoose S 1 , S 2 , . . . , S k ∈ S with replacement such thatS 1 ∪ S 2 ∪ · · · ∪ S k = U.The number of k-covers c k can be computed as followsInclusion Exclusion Formulac k = ∑ (−1) |X | s[X ] kX ⊆U
2/52Fast Exponential-Time AlgorithmsInclusion ExclusionComputing all s[X ]’s in time and space O ∗ (2 n )Build a table with 2 n entries containing s[X ] for all X ⊆ U. Thenevaluate the inclusion-exclusion formula in time O ∗ (2 n ).For instance, for every subset W ⊆ U disjoint from X , lets W [X ] denote the number of sets S ∈ S such that W ⊆ Sand S ∩ X = ∅.s W [X ] = s W [X ∪ {v}] + s W ∪{v} [X ] holds for all mutuallydisjoint X , W , and vcompute s[X ] = s ∅ [X ] recursively in time O ∗ (2 n ) and spaceO ∗ (2 n )
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Fast Exponential-Time Algorithms to solve NP-complete ... - Lita

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