Given a directed graph $G = (V,A)$, a feedback arc set is a set of arcs whose removal leaves an acyclic graph. The problem is to find the minimum cardinality such set.

I want to find out about is there some approximation algorithm around this problem.

  • 1
    $\begingroup$ What do you want to know about this problem? The corresponding decision problem is NP-complete. $\endgroup$
    – Shaull
    Sep 18, 2013 at 9:30
  • 2
    $\begingroup$ Then please edit your post and add details as to what you are looking for, and what you have already tried. You are likely to get more answers that way. $\endgroup$
    – Shaull
    Sep 18, 2013 at 9:37
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    $\begingroup$ If you are looking for an approximation algorithm, did you try googling for it? You'll find plenty of material. $\endgroup$
    – Juho
    Sep 18, 2013 at 19:55
  • $\begingroup$ @Shaull is there a fancy name for the problem? $\endgroup$
    – Subhayan
    Sep 19, 2013 at 2:54
  • $\begingroup$ "Feedback set" is the correct name. Fancy enough, I think. $\endgroup$
    – Shaull
    Sep 19, 2013 at 5:30

1 Answer 1


Kann's online compendium of NPO problems is a good place to start. Feedback Arc Set (the "Directed part is redundant when you use "arc") is:

  • APX-hard,
  • Approximable within $\mathcal{O}(\log n \log \log n)$ (where $n$ is the number of vertices).

The problem is also fixed-parameter tractable1, so it might make more sense to solve the problem exactly, rather than use what looks like a bad approximation algorithm. (Or as Pål points out below, the running time is a bit... unpleasant, so maybe not.)


1 - JACM publication, Razgon & Sullivan's preprint, Chen, Liu and Lu's preprint - the problem was independently solved and the results combined to one publication

  • $\begingroup$ Just because there's an FPT algorithm, doesn't really mean it makes more sense to solve the problem exactly. Especially not when the running time is $\Omega(n^4 k!)$. (Disclaimer: I have no idea what the best running time known for FAS is!) $\endgroup$
    – Pål GD
    Sep 19, 2013 at 8:47
  • $\begingroup$ BTW, is there anything like Kann's list that would be more up to date? Seems like the last update is from 2000. Probably not though... $\endgroup$
    – Juho
    Sep 19, 2013 at 13:38
  • $\begingroup$ I don't know of any similar resource unfortunately. It's a good starting point, but if you were serious, it's back to the more laborious search methods. $\endgroup$ Sep 20, 2013 at 0:00

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