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

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43/52Fast Exponential-Time AlgorithmsBranchingThe Algorithm of Fomin et al.System of Recurrences for the Main Branching RuleSystem of RecurrencesFor all m i and p h such that ∑ di=3 m i = dand ∑ dh=1 h p h = d + m 3 :P[k] ≤ P[k − ∆ OUT ] + P[k − ∆ IN ].Fortunately this can be restricted to a finite number of recurrencesby arguing that we may assume d ≤ 7.
4/52Fast Exponential-Time AlgorithmsBranchingThe Algorithm of Fomin et al.Optimizing the WeightsConsider an assignment of the weights w 3 , w 4 , w 5 , and w 6 whichsatisfies the initial assumptions and the constraints.It turns out that the number of subproblems that need to besolved for a problem of measure k is P[k] ≤ α k , where α > 1 isthe largest real root of the set of equationsα k = α k−∆ OUT+ α k−∆ IN.Thus the estimation of the running time reduces to choosing theweights w minimizing α = α(w) (we refer to Eppstein’s work onquasi-convex programming for a general treatment of such aproblem).
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