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

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1/52Fast Exponential-Time AlgorithmsBranchingThe Algorithm of Fomin et al.Analyzing the Main Branching Rule ISuppose we branch at a given node v, withN(v) = {u 1 , u 2 , . . . , u d } and d i = d(u i ).We denote by m i = m i (v) the number of nodes of degree i inN(v) :m i = |{u ∈ N(v) : d(u) = i}|.Let p h = p h (v) be the number of nodes in N 2 (v) which haveexactly h neighbors in N(v) :p h = |{w ∈ N 2 (v) : |N(w) ∩ N(v)| = h}|.Note that (at least) the nodes corresponding to p d−1 and p d aremirrors of v. Additionally, the number of edges between N(v) andN 2 (v) is p = p(v) = ∑ dh=1 h p h.We also define m ≥i = ∑ j≥i m j and p ≥h = ∑ j≥h p j.
2/52Fast Exponential-Time AlgorithmsBranchingThe Algorithm of Fomin et al.Analyzing the Main Branching Rule IILet P[k] be the maximum number of subproblems generated for aninstance of measure k.Reduction of the measure of the problem when discarding a vertexv :∆ OUT = ∆ OUT (v) ≥w d + p d w d + p d−1 w max{3, d−1} +d∑m i ∆ w i .Reduction of the measure of a problem when selecting a vertex v :w d +∆ IN = ∆ IN (v) ≥d∑m i w i + p 1 ∆ w d +i=3i=3d∑p h w max{3,h} .h=2
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Fast Exponential-Time Algorithms to solve NP-complete ... - Lita

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