## 1 Introduction  
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. For example, regarding the approximability of tree-width _tw_, it is known that there are polynomial time approximation algorithms with ratio \(O(\sqrt{\log{\mathit{tw}}(G)})\)[1]. Regarding inapproximability, tree-width cannot be approximated within any additive constant \(c\) unless P = NP [2]. Regarding rank-width _rw_, for every _fixed_\(k\), there is a polynomial time algorithm that reports \(\mathit{rw}(G)>k\), or outputs a rank decomposition of width at most \(3k-1\)[3]. Recently, it has been shown that maximum induced matching-width cannot be approximated within any constant factor in polynomial time unless NP = ZPP [4]. For several graph parameters, there are still large gaps between approximability and inapproximability results: it is a major concern as to whether there are constant factor approximation algorithms for those graph width parameters. Indeed, it is a long-standing open problem as to whether tree-width can be approximated within a constant factor.  
[PERSON] and [PERSON] introduced a complexity assumption referred to as the small set expansion hypothesis (SSEH) that is deeply related to the unique games conjecture (UGC) [5], and since then several inapproximability results under SSEH have been reported. For example, in [6] [PERSON] et al. showed that under SSEH there are no constant factor approximation algorithms for the balanced separator and minimum linear arrangement problems (a similar result was already known for the balanced separator problem under UGC [7]). Recently, [PERSON] showed inapproximability results for maximum biclique problems, minimum \(k\)-cut, and densest at-least-\(k\)-subgraph [8]. In [9], [PERSON] et al. (2014) showed under SSEH that there are no constant factor approximation algorithms for cut, path, tree-widths, minimum fill-in (it has recently been shown that minimum fill-in has no polynomial time approximation scheme unless P = NP and that assuming ETH, there is some positive \(\varepsilon\) such that no algorithm can find a \((1+\varepsilon)\) approximation in time \(2^{O(n^{1-\varepsilon})}\) for any positive constant \(\delta\)[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. However, the hardness result of tree-width in [9] does not necessarily mean that the long-standing open problem of tree-width is solved, because there is no consensus on the correctness of UGC and SSEH at this time [11] and it was shown in [12] that both unique games and small set expansion admit a subexponential time approximation algorithm. Regardless of the veracity of these two assumptions, the results in [6; 9] stimulate the study of approximation hardness for graph parameters, and it is widely acknowledged that UGC and SSEH have played important roles in the study of approximation algorithms.  
The above width parameters have widespread applications from both theoretical and practical viewpoints (e.g., [13]). Efficiently computing these width parameters becomes especially relevant when considering the fact that many NP-hard problems admit efficient graph algorithms for instances whose width parameters have a small value. The reader is invited to refer to the literature on _fixed parameter tractability_ for further information (e.g., [PERSON] and [PERSON] [14], [PERSON] et al. [15], [PERSON] and [PERSON] [16]). Inspired by [9], in this paper, we extend the research in [9] to other graph parameters. That is, we demonstrate in a unified manner that under SSEH there are no constant factor polynomial time approximation algorithms for cut, path, tree, branch, carving, NLC, rank, clique, boolean, and maximum induced matching-widths (see Figure 1 in Section 4).