This is a companion question to Results on number of solutions to random 3-SAT?
Let $A$ and $B$ be two problems drawn from random 3-SAT, both with the same number of variables and clauses. If $A$ has fewer satisfying assignments than $B$, do SAT solvers perform worse when trying to solve $A$ vs. $B$? Are there any results on this? (I'm a physicist by training, so a follow-up would be: is this even an interesting CS problem?)
EDIT: I wrote a quick little DPLL-based program to calculate the number of satisfying assignments for $\alpha = 4$ and $N = 80$. Here are the # of solutions and time to complete for a few random runs to illustrate the phenomenon that I'm looking for insight into:
- 368 solutions, 2.2 s
- 192 solutions, 8.4 s
- UNSAT, 3.2 s
- 50492 solutions, 2.6 s
- 1212 solutions, 13.9 s
As illustrated above, both the number of solutions and the time to run can fluctuate significantly; is there any relationship between the two?