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

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39/52Note that k = k(G) ≤ |V |, hence α k ≤ α n .Fast Exponential-Time AlgorithmsBranchingThe Algorithm of Fomin et al.Choice of the Measuren i := number of nodes of degree i in Gn ≥i := number of nodes of degree at least i in GTo analyze the maximum number of nodes of the search treegenerated for an instance G = (V , E), the following measurek = k(G) is used :Measurek(G) = ∑ i≥0w i n i ≤ nThe weights w i ∈ (0, 1] will be fixed such that the base of theexponential upper bound of the running time is minimized.[A computer is needed to do this.]
40/52Fast Exponential-Time AlgorithmsBranchingThe Algorithm of Fomin et al.Constraints and Linear RecurrencesAnalyzing the reduction rules (2) and (3)leads to constraints for the choice of the w i ’sDue to the domination rule (2) and the folding rule (3), when webranch, we can assume that, for every node w :(i) deg(w) ≥ 3 ;(ii) w is not dominated and does not dominate any other node.Analyzing the main branching rule (4)establishes a system of linear recurrences for any fixed choice ofthe w i ’s
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Fast Exponential-Time Algorithms to solve NP-complete ... - Lita

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