Figure 2. BSP algorithmic threshold on random 4-SAT problems.

The residual complexity per variable, Σ_res/N_res, goes to zero at the algorithmic threshold α_a^BSP. (a) The very small finite size effects, mostly producing a slight downward curvature at the right end, allow for a very reliable estimate of α_a^BSP via a linear fit. For random 4-SAT problems solved by BSP with r=0.9 we get α_a^BSP≈9.9, slightly beyond the rigidity threshold α_r=9.883(15), marked by a vertical line (the shaded area being its s.e.). (b) The same linear extrapolation holds for other values of r (red dotted line for r=0.5 and blue dashed line for r=0). The black line is the fit to r=0.9 data shown in panel (a). SID without backtracking (r=0) has a much lower algorithmic threshold, α_a^SID≈9.83. Error bars in both panels are the s.e.m.