What's the intuition for PTAS reductions instead of approximation preserving reductions or something of that ilk? I'd expect APX-Hardness to mean something like "finding a constant factor approximation for this problem is as hard as finding a constant factor approximation to any problem in APX" but having just a PTAS reduction from A to B isn't strong enough to imply a constant factor approx to B can be converted into a constant factor approx of A.


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Let me correct two misconceptions in your question:

  • $PTAS$-reductions are approximation-preserving

  • $APX$ is the class of problems that have constant-factor approximations

The big open question isn't whether $APX$ problems have constant-factor approximations (by definition, they do), but whether they have polynomial time approximation schemes. The meaning of $APX$-complete is "As hard as any problem in $APX$ to find a polynomial time approximation scheme for". A $PTAS$ from $A$ to $B$ reduction allows any polynomial time approximation scheme for $B$ to be used for $A$.


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