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 $\mathcal{F}_2(n,δn)$, the uniform distribution over all 2-CNFs with $δn$ clauses and $n$ variables, with $δ\geq 0$ a constant. We show that, with high probability, the random 2-CNF admits OBDDs of size polynomial in $n$ if $0 \leq δ< 1/2$ or if $δ> 1$. On the other hand, for $1/2 < δ< 1$, 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 $δ= 1/2$ and $δ= 1$. Both are known thresholds for other CNF properties, namely, $δ= 1$ is the satisfiability threshold for 2-CNF while $δ= 1/2$ 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.

💡 Research Summary

The paper investigates the size of Ordered Binary Decision Diagrams (OBDDs) required to represent random 2‑CNF formulas. The authors consider the uniform distribution F₂(n, δn) over all 2‑CNF formulas with exactly δn clauses on n variables, where δ ≥ 0 is a constant independent of n. Their main result is the existence of two sharp “compilability thresholds” at δ = ½ and δ = 1.

If δ < ½, then with probability tending to 1 as n → ∞, a random formula from F₂(n, δn) admits an OBDD of size at most n^c for some constant c. The proof rests on known results about the primal graph of a 2‑CNF: when δ < ½ the primal graph has treewidth at most 2 with high probability (a consequence of classic Erdős–Rényi results and the work of Lee, Lee, and Oum). Since pathwidth is at most O(treewidth·log n), the OBDD size, which is bounded by O(n·2^{pw})