Abstract
Wu et al. (2014) showed that under the small set expansion hypothesis (SSEH) there is no polynomial time approximation algorithm with any constant approximation factor for several graph width parameters, including tree-width, path-width, and cut-width (Wu et al. 2014). In this paper, we extend this line of research by exploring other graph width parameters: We obtain similar approximation hardness results under the SSEH for rank-width and maximum induced matching-width, while at the same time we show the approximation hardness of carving-width, clique-width, NLC-width, and boolean-width. We also give a simpler proof of the approximation hardness of tree-width, path-width, and cut-widththan that of Wu et al.
Highlights
There are many graph width parameters, such as cut, path, tree, band, branch, carving, clique, NLC, rank, boolean, maximum induced matching-widths, and the approximability and inapproximability of some of these width parameters have been investigated extensively
We briefly review the small set expansion hypothesis (SSEH), which is deeply related to the unique games conjecture (UGC)
Hardness Results Derived from Inapproximability of Tree-Width
Summary
There are many graph width parameters, such as cut, path, tree, band, branch, carving, clique, NLC, rank, boolean, maximum induced matching-widths, and the approximability and inapproximability of some of these width parameters have been investigated extensively. In [9], Wu et al (2014) showed under SSEH that there are no constant factor approximation algorithms for cut, path, tree-widths, minimum fill-in Algorithms 2018, 11, 173 δ [10]), one-shot black pebbling costs, and other problems Those were the first results showing the hardness of constant factor approximation for these graph parameters.
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