A polynomial-time solvable case for the NP-hard problem Cutwidth
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The Cutwidth problem is a notoriously hard problem, and its complexity is open on several interesting graph classes. Motivated by this fact we investigate the problem on superfragile graphs, a graph class on which the complexity of the Cutwidth problem is open. We give an algorithm that solves Cutwidth on superfragile graphs in O(n^2) time and O(n) space, thus resolving the complexity of the Cutwidth problem on superfragile graphs. We also explore the usefulness of the algorithm for cutwidth on threshold graphs by Heggernes, Lokshtanov, Mihai and Papadopoulos as an approximation algorithm for cutwidth on other graph classes. The Cutwidth problem is NP-hard for general graphs and a brute force algorithm would require O(n!) time. We give two faster algorithms solving the Cutwidth problem: One algorithm applying dynamic programming that runs in O^*(2^n) time and space, and one algorithm that runs in O^*(4^n) time and O(n\cdot log(n)) space by applying the divide-and-conquer technique. Finally we take a look at a similar problem called Optimal Linear Arrangement and suggest algorithms for solving the problem on threshold graphs and superfragile graphs in polynomial time.