Classification of OBDD size for monotone 2-CNFs

TL;DR

This paper introduces a new graph parameter, lu-mim width, and establishes tight bounds on the size of OBDDs for monotone 2-CNFs based on this parameter, linking graph theory and computational complexity.

Contribution

The paper defines lu-mim width and proves it tightly bounds the minimal OBDD size for monotone 2-CNF formulas, providing new insights into their complexity.

Findings

Bounds on OBDD size are tight and depend exponentially on lu-mim width.

Introduces the new graph parameter lu-mim width relevant to OBDD complexity.

Provides combinatorial results of independent interest related to the bounds.

Abstract

We introduce a new graph parameter called linear upper maximum induced matching width \textsc{lu-mim width}, denoted for a graph $G$ by $lu(G)$. We prove that the smallest size of the \textsc{obdd} for $\varphi$, the monotone 2-\textsc{cnf} corresponding to $G$, is sandwiched between $2^{lu(G)}$ and $n^{O(lu(G))}$. The upper bound is based on a combinatorial statement that might be of an independent interest. We show that the bounds in terms of this parameter are best possible.