Contraction and Treewidth Lower Bounds

Edge contraction is shown to be a useful mechanism to improve lower bound heuristics for treewidth. A successful lower bound for treewidth is the degeneracy: the maximum over all subgraphs of the minimum degree. The degeneracy is polynomial time computable. We introduce the notion of contraction degeneracy: the maximum over all graphs that can be obtained by contracting edges of the minimum degree. We show that the problem to compute the contraction degeneracy is NP-hard, but for fixed k, it is polynomial time decidable if a given graph G has contraction degeneracy at least k. Heuristics for computing the contraction degeneracy are proposed and experimentally evaluated. It is shown that these can lead to considerable improvements to the lower bound for treewidth. A similar study is made for the combination of contraction with Lucena’s lower bound based on Maximum Cardinality Search [12] .

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