The Compilability Thresholds of 2-CNF to OBDD

We prove the existence of two thresholds regarding the compilability of random 2-CNF formulas to OBDDs. The formulas are drawn from ${\mathcal{F}}_2(n,\delta n)$, the uniform distribution over all 2-CNFs with $\delta n$ clauses and $n$ variables, with $\delta \ge 0$ a constant. We show that, with high probability, the random 2-CNF admits OBDDs of size polynomial in $n$ if $0\le \delta <1\mathrm{/}2$ or if $\delta >1$. On the other hand, for $1\mathrm{/}2<\delta <1$, with high probability, the random $2$-CNF admits only OBDDs of size exponential in $n$. It is no coincidence that the two ``compilability thresholds’’ are $\delta =1\mathrm{/}2$ and $\delta =1$. Both are known thresholds for other CNF properties, namely, $\delta =1$ is the satisfiability threshold for 2-CNF while $\delta =1\mathrm{/}2$ is the treewidth threshold, i.e., the point where the treewidth of the primal graph jumps from constant to linear in $n$ with high probability.