How to sell my current solution to my advisor?

Question

Long story short, a long time ago (2 years) I had a research project, that for various reasons was not completed. Some months ago I came back to it and started trying new methods, state-of-the-art algorithms, some battle tested for decades, and some novel ones that came in the past few years (after the project got put in hiatus).

I don't disagree my method is naive, but it works fast and is more accurate than all the other ones. I need to mention, my method works only in a very, very niche subset of cases, all these other methods are designed to be general and work on arbitrary shapes, so my method outperforming other methods is not a great discovery it simply can leverage more assumptions than the other methods can.

I measured and tried a bunch of stuff and well, yes the method I came up with is really, really silly it's a naive approximation (it's definitely not an exact solution), but of all the other things I tried, this seems to solve this particular problem in this particular subdomain better than all the other algorithms I have used to solve the problem, based purely on the numbers.

If a paper comes out of this, the paper would essentially be "here's how you can leverage this silly math fact to come up with a rough approximation of the true solution in microseconds, vs hours or days for an exact one".

With that being shared, I don't know if or how I can pitch or sell this to my advisor, considering that after so much time I have come with a lot more work done, just to say that the method he didn't like the first time works better than the state-of-the-art algorithms (for this specific case).

Answer 1

How to sell my current solution to my advisor?

Just tell your supervisor what you've told us. (Using more positive language will help you.) Before you do: ask yourself whether the subdomain (for which your solution works) is interesting. Will others use your solution? (If not, your work might not be publishable, and your advisor might push back.) If so, you have advanced science in a meaningful way, congratulations, go publish.

Answer 2

In general trying to sell things to your advisor puts you on dangerous ground.

For example, the way you pitched your research in the question makes it seem like there are no problems that bar publication with your approach (as pointed out in the comments, fast approximations are generally publishable if they are novel). So I can't say why your advisor is hesitating. From the information given, it seems your advisor 1) is hesitating for no reason or 2) has a legitimate problem you didn't describe. If it's the 2nd case, be worried. Someone who brushes over and can't articulate legitimate problems with their approach is not someone who usually puts out high quality research. A big part of getting a PhD is learning how to be critical of your own work, and your advisor can't help you learn this if you don't listen to criticism.

Rather than sell to your advisor, I would open up more communication with them. Find out more about their reservations. Really try to get to the root of why the advisor doesn't want to publish. If your research has merit, that will come through in an honest investigation of its merits and its flaws. If your research has flaws, then listen to people who point them out, understand the flaws, and work on figuring out how to fix them.