Solving a corn puzzle with CP-SAT
thill.me
thill.me
As a kid, it seemed obvious to me that a good model was the digits 1 through 8, representing the row that a queen would be in. Doing a recursive backtracker on an ordering of the digits and testing each pair for diagonal (digit i minus digit j was equal to +/- i-j) and my BASIC program spit out the solutions in twenty minutes.
I didnt enter the contest because I figured everybody would do that. Turns out when the editor published the 'best' solutions, they all, every single one, modeled the board as an 8 by 8 array and ran around following diagonals iteratively. And took hours to complete.
Anyway, choosing a good solution model is most of the problem solved.
You still need to get the backtracking correct and there's some minor tricks to work out, but that'll get you up to n=8 working quite well.
The model (8 different digits representing row in that column) no queen was in the same row or column, by construction. So only the diagonal test remains.
If this was a thousand piece puzzle, I would still venture recursive backtracker with good heuristics will beat CP-SAT, even in the sudoku case some good heuristics with backtracking beats CP-SAT. Not sure why Claude immediately jumped to using CP-SAT.
I've spent significant chunks of my career help people throw away backtracking searchers people polished over years with a CP-SAT model I threw together in 30 minutes, often much to their upset.
You can for Sudoku often beat a CP-SAT solver, but that's because the problems are trivial and take milliseconds. If you look at more difficult Sudoku variants, or 16x16 grids, backtracking solvers start to fall behind.
This is one that is hard for people to really internalize, I think? The SAT solvers many are likely to use today are not at all the same as the ones they would have used 20 years ago. They have made some amazing advances in how to approach those problems.
There are also probably some very poorly conceived models that people use to adapt a problem to some of these solvers.
Do you have examples that are public or that you can talk about?
With one of these solver-based approaches, you encode the problem with decision variables in some fashion, state all the constraints & call solve. There's some art & experience in how to encode & model the problem, but the specification of the model & the problem is fairly declarative.
What's great about general purpose solver-based approaches is that its usually faster (in terms of implementation time & effort) to start getting solutions & they're also much more robust to changes in requirements & the problem statement. That's less of a concern in this toy example, where the problem is small, well-defined & unambiguous, but in a real business/industrial application, the problem statement often changes considerably over time.
I agree that the performance & behaviour of a black box solver may be much harder to reason about than something you custom build by hand & know inside out, but if the general purpose black box solver is 'good enough' for the distributions of problem instances it needs to process, then there's no need to custom-build anything. Throw the black box solver at it -- job's done, and you're left with something that's both quite readable (declarative modelling of the problem, particularly if someone documents the formulation - the meaning of all the decision variables, index sets, constraints, objective terms etc) & flexible to future change.
Custom solvers & heuristics can sometimes be much, much more effective in being able to scale and solve industrial-scale problems, but usually at the expense of being much more effort to set up in the first place, and very fragile to changes in requirements -- if you learn something a few weeks into a project that perturbs the problem statement, maybe it wrecks the particular mathematical structure you were relying upon for a custom solver/heuristic, so you need to chuck out all your work & go back to the drawing board.
"Feed it to SAT solver" is a super useful technique every seasoned programmer should have in their toolbelt.
If you want to improve your programming skill, I suggest implementing your own version of these two interesting, satisfying algorithms:
- DPLL, the classic general-purpose SAT solving algorithm [3].
- Algorithm X (aka Dancing Links), Knuth's super elegant search algorithm for solving exact cover problems [4] [5].
You're not going to build something that beats CaDiCaL [6] in an afternoon, but writing your own implementation of these algorithms will teach you a lot, and a reasonable time investment will get you to a pretty satisfying endpoint of having a practical solver for your favorite application (Sudoku, polyomino packing, etc.)
[1] Reinventing wheels is not a bad way to spend your days at a retreat focusing on one's personal growth as a programmer. "I think I'll learn a lot and get some good practice at programming and algorithmic thinking" is an easy bar to meet.
In a professional context, telling your company or client they should commit scarce, expensive engineering resources (e.g. your time) to creating and maintaining their own wheel design requires justification [2].
[2] "Don't reinvent wheels" is good general advice for junior developers. Senior developers know that sometimes there are specific situations where this general advice doesn't apply. For example, if no existing wheels meet your application's requirements, or if you have ideas for proprietary features that give you a competitive advantage. Even in a professional setting, reinventing the wheel isn't always a bad idea; you just need to meet a much higher bar to justify the commitment of resources.
[3] https://en.wikipedia.org/wiki/DPLL_algorithm
[4] https://arxiv.org/abs/cs/0011047
If you want to try to implement one yourself, a simple one is Simulated Annealing [1]. The core algorithm shouldn't be more than 30 lines of code; it is simple to implement. If you want to try an existing library, you can try Timefold Solver [2] (disclosure: I work for Timefold).
I'm not familiar with CP-SAT, but TTBOMK all SAT solvers use a type of backtracking search underneath called DPLL. Modern ones are highly tuned in terms of which variable they choose to branch on next, and in what order to try its possible values; this can have an enormous impact on runtime. They probably use several tricks on top of that; the big one that I'm aware is conflict-driven clause learning, where the solver adds new constraints that it discovers as it goes along (e.g., it might be able to determine that x and y always have the same value in every solution), which can shrink the search space a lot.
https://en.wikipedia.org/wiki/Constraint_satisfaction_proble...
It just evaluates the data and constraints to search the more obvious paths first, and cut illegal paths earlier. And some integer tricks, and a cache of learned constraints. Which the Russians found, when trying to solve Chess efficiently. IBM/Ken Thompson just threw hardware at it, while the Russians made the algorithmic advances. Just as now with LLM's. The US folks just throw hardware at it, whilst the Chinese optimize their algos.
Eventually shitting on LLM code will be seen by all as lazy cope. I too am aware of the existence of SAT but I really would struggle to immediately see through some problem i was having and interpret SAT unless I did it a bunch. Having agent suggest the "right thing" is clearly better. And hopefully would help my intuition in the future.