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I am currently facing a situation that I am not sure how to handle.

Basically, there is this problem $A$. Problem $A$ has been a hot topic in the past $5$ years. Last year, a paper was published in a very reputable journal (IEEE Transactions on Information Theory) that proposed a new algorithm that consists of $2$ sub-algorithms, $A_1, A_2$ that are both used to solve the problem. Here is the caveat, Algorithm $A_1$ was shown to eventually converge without any run time gurantee (shown to converge with finite number of steps), while $A_2$ was shown to take $O(n^5)$.

My paper is essentially replacing $A_1$ with a new algorithm which is guaranteed to terminate correctly in $O(n\log n)$, hence bringing the overall time complexity when combined with $A_2$ to $O(n^6\log n)$.

So far, so good. I am fairly certain the paper can get accepted into the same journal since the new algorithm is highly non trivial and provides lots of new insight on the problem. However, 2 weeks ago, I was wrapping up everything and decided to give algorithm $A_2$ another read as well as its proofs. That's when I faced some problems.

One of the proofs in that paper is faulty, beyond repair. The author did a very big jump, and I managed to come up with a counter example essentially showing that the theorem he "proved" is false. I tried to fix his algorithm, but I couldn't. So I tried approaching it from a new perspective using his method, and I managed to get a new algorithm that replaces the faulty $A_2$, but it ran in $O(n^6)$.

Here is my problem, my advisor thinks I should definitely include this in my paper as I've essentially replaced both algorithms $A_1, A_2$. However, If I just say I have an $O(n^6)$ when there is an $O(n^5)$ known, its going to raise eye brows. The only way I can pass this is if I actually take jabs at $A_2$ and show that its actually wrong, but I feel that "attacking" a paper in my first publication is a really bad idea.

Any ideas?

EDIT: Contacted the author and he acknowledged the mistake. He agreed to be mentioned in the new paper and he showed quite the interest in the $O(n\log n)$ algorithm. He also contacted IEEE Transactions to see if they can add a correcting note. This went way smoother than I thought it would. Thanks everyone!

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    $\begingroup$ Finding and fixing a flaw in a paper is not a bad thing and a good scholar would not consider it an "attack", but rather the normal process of research $\endgroup$ – Max New Sep 21 '18 at 18:40
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    $\begingroup$ It's not bad at all, people do it all the time and nobody gets offended if you point out that the algorithm is wrong (if it indeed is). $\endgroup$ – domotorp Sep 21 '18 at 18:40
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    $\begingroup$ Seconding the two comments above; with a small caveat. Phrasing is important. Saying "However, there appears to be a flaw in a key lemma, and the stated runtime guarantee does not follow from the analysis; it is unclear if it holds." is reasonable; saying "The authors' proof is wrong and beyond repair, and their algorithm does not work." may be less of a good idea. $\endgroup$ – Clement C. Sep 21 '18 at 21:36
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    $\begingroup$ You should also consider contacting the author privately before submitting the paper and/or making it public. Although it sounds like you and your advisor are confident that there is really a counterexample to one of the author's claims and that the original algorithm is flawed, there is a small chance that you are misunderstanding the argument and/or that there is a simple workaround you are overlooking. Being in clear communication with the original author will make your life easier. $\endgroup$ – Noam Zeilberger Sep 22 '18 at 11:42
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    $\begingroup$ @user3508551: Here's a secret -- getting a polite note from a graduate student saying that your theorem is broken actually feels, on net, good. Yes, of course you feel sad to learn you made a mistake and your theorem is wrong, but on the other hand, getting concrete evidence that someone smart spent a lot of time reading and thinking about your work is very rewarding. $\endgroup$ – Neel Krishnaswami Sep 24 '18 at 9:07
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I think the same standards apply, regardless of whether it's your 1st or 100th publication. If you think you've found a mistake in a published paper, a common courtesy is to first contact the paper's authors for a clarification, as Noam suggested in the comments. If the authors confirm that it's indeed a mistake, you can indicate that in the paper (cite it as "private communication"); this should make the review process smoother.

When pointing out people's mistakes in print, gentle understatement is a good practice. "There appears to be a mistake in the proof" should be gentle enough. Having a counterexample certainly helps.

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    $\begingroup$ Contacted the author and he acknowledged the mistake. He agreed to be mentioned in the new paper and he showed quite the interest in the $O(n\log n)$ algorithm. He also contacted IEEE Transactions to see if they can add a correcting note. This went way smoother than I thought it would. Thanks you! $\endgroup$ – user3508551 Sep 23 '18 at 17:17
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    $\begingroup$ That's great news! Hopefully the paper will be accepted soon -- when it is, maybe you'll post a link here so we can read it :) $\endgroup$ – Aryeh Sep 24 '18 at 18:29

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