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Meaning of L-reduction from Dominating set problem

MathOverflow Asked by Venugopal K on August 7, 2020

We are working in a variation of Locating dominating sets. Recently, we realized that the reduction from dominating set to our problem in proving its NP-completeness turns out to be also an L-reduction. This would mean that there is no PTAS for our problem since the dominating set problem does not have PTAS. Now we have two questions:

  1. since the dominating set problem is W[2]-complete, does that follow to our problem because of the L-reduction?
  2. we have proved that our problem is in XP (i.e, piecewise polynomial) and also in APX (there is a 2-approximation polynomial algorithm). But dominating set problem is known to be not in APX. Does this mean somewhere we might have gone wrong or is this perfectly alright?

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