The Propagation Depth of Local Consistency
This provides foundational insights into the inherent sequential nature of local consistency algorithms, impacting constraint satisfaction and parallel computing, but is incremental as it extends known bounds for specific k to a general result.
The paper tackled the problem of determining the minimum number of nested propagation steps required by k-consistency algorithms on binary constraint networks, establishing optimal bounds of Ω(n^{k-1}d^{k-1}) steps for k ≥ 2, which is tight as the overall steps are at most n^{k-1}d^{k-1}.
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 $Ω(nd)$ and 3-consistency requires $Ω(n^2)$ 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 $Ω(n^{k-1}d^{k-1})$ 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 $n^{k-1}d^{k-1}$.