Abstract
Tree-width and path-width are widely successful concepts. Many NP-hard problems have efficient solutions when restricted to graphs of bounded tree-width. Many efficient algorithms are based on a tree decomposition. Sometimes the more restricted path decomposition is required. The bottleneck for such algorithms is often the computation of the width and a corresponding tree or path decomposition. For graphs with n vertices and tree-width or path-width k, the standard linear time algorithm to compute these decompositions dates back to 1996. Its running time is linear in n and exponential in \(k^3\) and not usable in practice. Here we present a more efficient algorithm to compute the path-width and provide a path decomposition. Its running time is \(2^{O(k^2)} n\). In the classical algorithm of Bodlaender and Kloks, the path decomposition is computed from a tree decomposition. Here, an optimal path decomposition is computed from a path decomposition of about twice the width. The latter is computed from a constant factor smaller graph.
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