My first thought is that there is surely an Integer Linear Programming approach that can solve this within a few seconds using some very advanced solver like Gurobi.
These solvers can very often find the globally optimal solution - and prove that this solution is optimal - much faster than exhaustive search.
Of course they do use a smart exhaustive search by applying branch-and-bound as described in this article, but advanced solvers use, among other things, branch-and-cut where cutting planes in the solution space are generated, as well as a variety of other approaches.
One interesting thing however is that GPUs are still not particularly applicable for solvings Mixed Integer Linear Programs to sufficient accuracy. There are things like PDLP that can use GPUs to solve these problems, but they are restricted to something like 1e-4 accuracy whereas the state of the art is more like 1e-9.
I tried Z3 and OR Tools. I didn't try Gurobi. But this was enough to make me think ILP was a dead end. (There were a lot of dead ends in this project.)
I don't know much about integer programming, though, and I'd love to be proven wrong.
I saw that! In my experience, problems that seem completely intractable using open source tools often get solved in seconds using state of the art commercial approaches.
One interesting thing about Boggle is that the number of variables (16 cells) is very small compared to the number of coefficients on how they combine (the number of possible words).
Simulated annealing [1] is mentioned but not explained in the list of modifications to hill climbing. The technique roughly is: accept modifications to the board which decrease the score, with a probability inversely related to the magnitude of the decrease, and which decreases as the search progresses. This helps avoid getting stuck in local maximae.
Annealing is mentioned a few times in the post but not discussed in any detail. I found that hill climbing with an expanded "pool" of boards and exhaustive search of neighbors was the most reliable way to get from a random starting point to the highest-scoring board: https://github.com/danvk/hybrid-boggle/blob/main/boggle/hill...
The article actually does mention using this technique, though it doesn't explain it, so thanks for the background from someone who isn't familiar with this space!
My first thought is that there is surely an Integer Linear Programming approach that can solve this within a few seconds using some very advanced solver like Gurobi.
These solvers can very often find the globally optimal solution - and prove that this solution is optimal - much faster than exhaustive search.
Of course they do use a smart exhaustive search by applying branch-and-bound as described in this article, but advanced solvers use, among other things, branch-and-cut where cutting planes in the solution space are generated, as well as a variety of other approaches.
One interesting thing however is that GPUs are still not particularly applicable for solvings Mixed Integer Linear Programs to sufficient accuracy. There are things like PDLP that can use GPUs to solve these problems, but they are restricted to something like 1e-4 accuracy whereas the state of the art is more like 1e-9.
I actually did try ILP, see https://stackoverflow.com/questions/79422270/why-is-my-z3-an...
I tried Z3 and OR Tools. I didn't try Gurobi. But this was enough to make me think ILP was a dead end. (There were a lot of dead ends in this project.)
I don't know much about integer programming, though, and I'd love to be proven wrong.
I saw that! In my experience, problems that seem completely intractable using open source tools often get solved in seconds using state of the art commercial approaches.
If you want to give it a try, I'd love to hear if that's the case! It's deleted in the repo now, but here's code to generate a spec for an ILP solver: https://github.com/danvk/hybrid-boggle/blob/62d3f01aed802734...
One interesting thing about Boggle is that the number of variables (16 cells) is very small compared to the number of coefficients on how they combine (the number of possible words).
I am very intrigued by this. I’ll do something thinking this evening about how a tight Boggle model may look.
Great! Feel free to reach out -- my email isn't hard to find.
Simulated annealing [1] is mentioned but not explained in the list of modifications to hill climbing. The technique roughly is: accept modifications to the board which decrease the score, with a probability inversely related to the magnitude of the decrease, and which decreases as the search progresses. This helps avoid getting stuck in local maximae.
[1] https://en.wikipedia.org/wiki/Simulated_annealing
EDIT: Somehow I didn't see that simulated annealing was mentioned by name (but not explained), ha!
Annealing is mentioned a few times in the post but not discussed in any detail. I found that hill climbing with an expanded "pool" of boards and exhaustive search of neighbors was the most reliable way to get from a random starting point to the highest-scoring board: https://github.com/danvk/hybrid-boggle/blob/main/boggle/hill...
Somehow my eyes missed that! Edited my comment.
The article actually does mention using this technique, though it doesn't explain it, so thanks for the background from someone who isn't familiar with this space!
The article does indeed mention simulated annealing though?
Somehow I didn't see that, good catch! (It's mentioned but not explained.) Edited my comment.