On oblivious branching programs with bounded repetition that cannot efficiently compute CNFs of bounded treewidth

TL;DR

This paper investigates the limitations of bounded-repetition oblivious branching programs, specifically c-OBDDs, in efficiently computing CNFs with bounded treewidth, establishing size lower bounds and model separations.

Contribution

It introduces size lower bounds for c-OBDDs on CNFs of bounded treewidth and demonstrates a separation from sentential decision diagrams, advancing understanding of computational complexity in this domain.

Findings

Lower bounds of size Ω(n^{k/(8c-4)}) for c-OBDDs on certain CNFs

Separation between c-OBDDs and SDDs based on these bounds

Matching width and pathwidth are linearly related

Abstract

In this paper we study complexity of an extension of ordered binary decision diagrams (OBDDs) called $c$-OBDDs on CNFs of bounded (primal graph) treewidth. In particular, we show that for each $k$ there is a class of CNFs of treewidth $k \geq 3$ for which the equivalent $c$-OBDDs are of size $\Omega(n^{k/(8c-4)})$. Moreover, this lower bound holds if $c$-OBDD is non-deterministic and semantic. Our second result uses the above lower bound to separate the above model from sentential decision diagrams (SDDs). In order to obtain the lower bound, we use a structural graph parameter called matching width. Our third result shows that matching width and pathwidth are linearly related.