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Comment by laserbeam | Hacker News Reader

I honestly think that's just journalism for "no one implemented it in production yet". Which is not surprising, for an algorithm less than a year old. I don't think it's worth expanding and explaining "too much work".

That being said, sometimes if an algorithm isn't the fastest but it's fast and cheap enough, it is hard to argue to spend money on replacing it. Which just means that will happen later.

Furthermore, you might not even see improvements until you implement an optimized verision of a new algorithm. Even if big O notation says it scales better... The old version may be optimized to use memory efficiently, to make good use of SIMD or other low level techniques. Sometimes getting an optimized implementation of a new algorithm takes time.

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As other commenters here have mentioned, in discrete optimization there can be a very large gap between efficienct in theory and efficient in practice, and it is very likely that this is the case here too. Linear programming for example is known to be solvable in polynomial time, but the algorithm which does so (the ellipsoid method) is not used in practice because it is prohibitively slow. Instead, people use the (exponential time worst-case) simplex method.

Modern ILP solvers have a huge number of heuristics and engineering in them, and it is really difficult to beat them in practice after they have optimized their branch-and-cut codes for 30 years. As the top comment mentions, the software improvements alone are estimated to have improved the solving time of practical ILP instances by a factor of 870'000 since 1990.

I thought there were other interior point methods now beside the ellipsoid algorithm that performed better. Some of these are useful in convex nonlinear programming, and I believe one is used (with a code generator from Stanford to make it faster) in the guidance software for landing the Falcon 9 first stage. There, as the stage descends it repeatedly solves the problem of reaching the landing point at zero velocity with minimum fuel use, subject to various constraints.

Yes, there are other interior point methods besides the ellipsoid method, and virtually all of them perform better for linear programming. Sometimes, the solvers will use these at the root node for very large models, as they can beat out the simplex algorithm. However, I am unsure if any of them has been proven to run in polynomial time, and if so, if the proof is significantly different from the proof for the ellipsoid method. The point I was mainly trying to make is that there can be a significant gap between practice and theory for ILP. Even 40 years after LP was proven to be polytime solvable, simplex remains the most widely used method, and it is very hard for other methods to catch up.

Karmarkar's algorithm, for example, has been proved to run in polynomial time.

https://en.wikipedia.org/wiki/Karmarkar%27s_algorithm

It was also (in)famous as an algorithm that was patented (the patent expired in 2006).

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