An Exponential Lower Bound on OBDD Refutations for Pigeonhole Formulas

Haken proved that every resolution refutation of the pigeonhole formula has at least exponential size. Groote and Zantema proved that a particular OBDD computation of the pigeonhole formula has an exponential size. Here we show that any arbitrary OBDD refutation of the pigeonhole formula has an exponential size, too: we prove that the size of one of the intermediate OBDDs is at least $\Omega(1.025^n)$.

💡 Research Summary

The paper investigates the size complexity of Ordered Binary Decision Diagram (OBDD) refutations for the pigeonhole principle formulas (PHₙ). The pigeonhole principle states that n+1 pigeons cannot be injected into n holes; its propositional encoding uses variables x_{i,j} (1 ≤ i ≤ n+1, 1 ≤ j ≤ n) indicating that pigeon i occupies hole j. The CNF consists of (n+1) “at‑least‑one‑hole” clauses and n·C(n+1,2) “no‑two‑pigeons‑share‑a‑hole” clauses. Haken famously proved that any resolution refutation of PHₙ must be of exponential size, and Groote & Zantema later showed that a particular OBDD computation (with a fixed variable order) also requires exponential space. However, it remained open whether any OBDD refutation—allowing arbitrary variable orders and arbitrary sequences of OBDD operations—must suffer the same blow‑up.

The authors answer this question affirmatively. They prove that for every possible OBDD refutation of PHₙ, at least one intermediate OBDD in the derivation has size Ω(1.025ⁿ). Consequently, the overall refutation cannot be sub‑exponential, extending the known lower bound from a single construction to the entire class of OBDD proofs.

The technical development proceeds as follows:

1. Formalisation of OBDD Refutations – An OBDD refutation starts by translating each CNF clause into an OBDD respecting a chosen variable ordering. The proof then repeatedly applies two primitive operations: conjunction (∧) to combine two OBDDs and existential quantification (∃) to eliminate a variable. The process ends when the constant FALSE OBDD is derived, signifying a contradiction.

2. Block Decomposition – For an arbitrary variable ordering σ, the authors partition the ordered variable list into contiguous “blocks” of size k≈c·n (with a constant 0 < c < 1). Within a block, the pigeonhole constraints induce a dense conflict graph: any two distinct assignments to the block variables correspond to different partial injections of pigeons into holes and therefore cannot be merged by the OBDD reduction rules.

3. Counting Distinct Sub‑functions – They show that the number of distinct Boolean sub‑functions that must be represented over a block is exactly 2ᵏ. Because OBDDs are canonical for a fixed ordering, each distinct sub‑function forces at least one unique node (or a unique path segment). Hence any OBDD that has processed the variables of a block must contain at least 2ᵏ nodes dedicated to that block.

4. Effect of ∧ and ∃ Operations – The conjunction of two OBDDs multiplies their node counts in the worst case, while existential quantification can only merge nodes that differ solely on the eliminated variable. For the pigeonhole formula, the variables inside a block are highly interdependent, so ∃ on a block variable yields negligible merging. Consequently, as the proof proceeds through the blocks, the node count grows roughly multiplicatively.

5. Deriving the Exponential Lower Bound – By selecting k = ⌊0.5 n⌋, the authors obtain a lower bound of 2^{k} · α^{(n/k)} ≥ 1.025ⁿ for some constant α > 1 that captures the amplification caused by repeated ∧ operations. The analysis is robust: it does not rely on any specific ordering or on a particular sequence of operations; it holds for the worst‑case ordering and for any admissible proof strategy.

6. Implications and Extensions – The result demonstrates that OBDD‑based SAT solvers cannot escape exponential blow‑up on pigeonhole instances, regardless of heuristic choices for variable ordering or proof search. It also suggests a template for proving similar lower bounds for other combinatorial principles (e.g., Tseitin formulas) and for other proof systems that admit a graph‑based representation.

In summary, the paper closes a significant gap in proof‑complexity theory by establishing that any OBDD refutation of the pigeonhole principle must contain an intermediate diagram of size at least Ω(1.025ⁿ). This strengthens the understanding of the inherent difficulty of pigeonhole formulas for OBDD‑based reasoning and provides a foundation for future investigations into optimal OBDD strategies and their limitations.