A Moderately Exponential Time Algorithm for k-IBDD Satisfiability

Abstract

A k-indexed Binary Decision Diagram (k-IBDD) is a branching program with k-layers and each layer consists of an Ordered Binary Decision Diagram (OBDD). This paper studies the satisfiability of k-IBDD (k-IBDD SAT). A k-IBDD SAT is, given a k-IBDD, to ask whether there exists a consistent path from the root to the 1-sink. We propose a moderately exponential time algorithm using exponential space for k-IBDD SAT of n variables and cn size. Our algorithm runs in time $O\left( 2^{(1-\mu (c))n}\right)$, where $\mu (c)=\Omega \left( \frac{1}{(\log {c})^{2^{k-1}-1}}\right)$. As a corollary, we obtain a polynomial space and deterministic algorithm, which solves k-IBDD SAT of size polynomial in n and runs in $O\left( 2^{ n - n^{ 1/2^{k-1} }}\right)$ time.