Comment by naet
6 months ago
I actually wrote a backtracking solution to the LinkedIn queens game a while ago (and the tango game).
I know nothing about SMT or SAT and I imagine they might be faster, but the backtracking appears to solve just as instantaneously when you push the button.
Might be cool to learn a bit about SMT or SAT. Not sure how broadly applicable they are but I've seen people use them to solve some difficult advent of code problems if nothing else.
SAT (satisfiability), SMT (satisfiability modulo theories), CP (constraint programming) and so on are not magic solutions. They are more or less really smart backtracking solvers that use lots of heuristics, deductions based on the constraints, learning, restarts, portfolios, and so on. A real benefit is that one can re-use the algorithms that someone else has developed and debugged.
For toy-sized problems such as the LinkedIn Queens, 8-queens, and 9x9 Sudoku more or less any competently implemented solver will work fine. As soon as the size of the problem starts to grow, the difference between a plain backtracking algorithm and a SAT/SMT/CP system.
Definite broad applicability.
NP-complete are the hardest problems in NP. Cook in 1971 proved SAT to be in NP-complete. In the worst case for any other problem in NP, we can quickly (i.e., in polynomial time) convert instances of that problem into instances of SAT. In other words, we can use SAT to solve any problem in NP.
It turns out there are many problems in NP-complete. The fast conversion applies among them too, so in some sense, problems in NP-complete are all the same because we can use them all to solve instances of each other. However, for some of those problem instances the best known algorithm is to try all possible inputs, which requires exponential time (very, very slow for even modestly large inputs).
Lots of research has been and continues to be poured into SAT because any gains automatically yield improvements to everything else in NP-complete and the rest of NP. Using a SAT solver allows you to hitch a ride more or less for free on the results of all that research. Each incremental improvement to SAT solvers benefits programs that use them.
As the author noted, forming SAT instances by hand can be a pain. SMT or SAT Modulo Theories is sort of a high-level language that “compiles down” to SAT. Expressing problems with SMT is more natural and reduces the burden of converting your problem to SMT and SMT solutions back to your problem domain.