The Propagation Depth of Local Consistency

TL;DR

This paper establishes tight bounds on the minimum number of nested propagation steps required by $k$-consistency algorithms in binary constraint networks, revealing their inherent sequential nature and limits on parallelization.

Contribution

It provides the first exhaustive characterization of the minimal nested propagation steps needed for $k$-consistency, showing these bounds are tight and depend on network size and domain.

Findings

For $k extgreater=2$, the minimum nested steps are $ heta(n^{k-1}d^{k-1})$.

The bounds are tight, matching the maximum number of steps $k$-consistency algorithms perform.

Local consistency algorithms are inherently sequential, limiting parallelization.

Abstract

We establish optimal bounds on the number of nested propagation steps in $k$-consistency tests. It is known that local consistency algorithms such as arc-, path- and $k$-consistency are not efficiently parallelizable. Their inherent sequential nature is caused by long chains of nested propagation steps, which cannot be executed in parallel. This motivates the question "What is the minimum number of nested propagation steps that have to be performed by $k$-consistency algorithms on (binary) constraint networks with $n$ variables and domain size $d$?" It was known before that 2-consistency requires $\Omega(nd)$ and 3-consistency requires $\Omega(n^2)$ sequential propagation steps. We answer the question exhaustively for every $k\geq 2$: there are binary constraint networks where any $k$-consistency procedure has to perform $\Omega(n^{k-1}d^{k-1})$ nested propagation steps before local inconsistencies were detected. This bound is tight, because the overall number of propagation steps performed by $k$-consistency is at most $n^{k-1}d^{k-1}$.