The Propagation Depth of Local Consistency

  • Christoph Berkholz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8656)


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 ≥ 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.


Propagation Step Critical Position Winning Strategy Constraint Network Local Consistency 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Christoph Berkholz
    • 1
  1. 1.RWTH Aachen UniversityAachenGermany

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