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Monday, September 26, 2022

Is the complexity of approximating Vertex Cover of degree 3 open? (ADDED LATER-NO)

 RECALL:

A max-problem f has a A P Time App Scheme (PTAS) if there is an algorithm ALG such that

 ALG(\epsilon) \ge (1-\epsilon)f(x).


A min-problem f has a A P Time App Scheme (PTAS) if there is an algorithm ALG such that

 ALG(\epsilon) \le (1+\epsilon)f(x).


(Note that the poly can depend on epsilon so it may be something like n^{1/epsilon}.)


MAX3SAT is, given a formula with \le 3 literals per clause, find an assignment

that maximized the number of clauses satisfied.


VCB-a is Vertex cover where graphs have degree \le a


The following are known:

0) MAX3SAT is in APX.

1) The PCP paper, here, showed that if MAX3SAT has a PTAS then P=NP.

2) Papadimitriou and Yannakakis (here)  had showed much earlier that MAX3SAT \le VCB-4 with an approx preserving reduction.

3) From (1) and (2) we have that VCB-4 has a PTAS then P=NP. (VC is in APX by an easy 2-approx).

4) Clearly VCB-2 is in P.

The following seems to be open, though if you know otherwise pleae leave a comment:


Is VCB-3 a) in P? b) NPC? (ADDED LATER- NPC- See comments.) 

Is the following true: if VCB-3 has a PTAS then P=NP. (ADDED LATER- NO PTAS-See Comments)


NOTE- all of the above is true for Ind Set-4 and Dom Set-4. So that leads to more open problems.


6 comments:

  1. Vertex Cover on graphs of degree 3 is NP-Complete. This was proven in 1974 by Garey, Johnson and Stockmeyer in the paper "Some simplified NP-complete problems", in their Theorem 2.4.

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    Replies
    1. Great, thanks. I think the PTAS problem really is open but would be happy to be proven wrong.

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  2. If one begins the reduction of the above-mentioned Thm 2.4 from Max 3SAT where each variable occurs a bounded number of times, the same reduction probably establishes APX-hardness with a more careful soundness argument?

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  3. The problem is APX-hard, as proved in https://doi.org/10.1016/S0304-3975(98)00158-3

    A more detailed, but not updated, discussion is at https://www.csc.kth.se/~viggo/wwwcompendium/node10.html

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  4. THANKS to both Venkat and Anonymous for the references! Google will never replace asking around!

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  5. This post - and another recent one - have made me think you should consider using MathJax so that the LaTeX you use in your posts can actually render attractively, instead of requiring the reader to parse it

    This article makes it look very easy to do.
    https://www.airinaa.com/2018/04/add-mathjax-beautiful-mathematic.html

    ReplyDelete