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I am confused whether $\mathsf{APX-hard} \subseteq \mathsf{NP-hard}$. My confusion stems from a result on graph pricing from this paper which says the following : "Unlike the general case of the graph pricing problem, the bipartite case was not even known to be NP-hard. We show that it is in fact APX-hard by a reduction from MAX CUT". It seems like the authors are implying that APX-hardness is a stronger property than NP-hardness

Since $\textsf{APX} \subseteq \textsf{NP}$ by definition, the above statement is true for MAXCUT as it is in $\textsf{APX}$ and also in $\textsf{APX-hard}$ but I am not sure if $\mathsf{APX-hard} \subseteq \mathsf{NP-hard}$ in general. Any pointers/counter example problem is appreciated!

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    $\begingroup$ What the authors mean is the following. Previously optimization problem X was not known to be NP-Hard; it could in fact be in P. So the first step is to prove NP-Hardness. If a problem is NP-Hard it could have a PTAS or even an FPTAS. The authors rule this out by showing that there is a constant c > 1 such that approximating to within a factor of c is also NP-Hard. That is all they are saying. $\endgroup$ – Chandra Chekuri Jan 22 '18 at 2:18

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