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The Complexity of the Distributed Constraint Satisfaction Problem

Authors Silvia Butti , Victor Dalmau

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Silvia Butti
  • Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Spain
Victor Dalmau
  • Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Spain


We would like to thank Gergely Neu for useful discussions on the weighted majority algorithm.

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Silvia Butti and Victor Dalmau. The Complexity of the Distributed Constraint Satisfaction Problem. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 20:1-20:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


We study the complexity of the Distributed Constraint Satisfaction Problem (DCSP) on a synchronous, anonymous network from a theoretical standpoint. In this setting, variables and constraints are controlled by agents which communicate with each other by sending messages through fixed communication channels. Our results endorse the well-known fact from classical CSPs that the complexity of fixed-template computational problems depends on the template’s invariance under certain operations. Specifically, we show that DCSP(Γ) is polynomial-time tractable if and only if Γ is invariant under symmetric polymorphisms of all arities. Otherwise, there are no algorithms that solve DCSP(Γ) in finite time. We also show that the same condition holds for the search variant of DCSP. Collaterally, our results unveil a feature of the processes' neighbourhood in a distributed network, its iterated degree, which plays a major role in the analysis. We explore this notion establishing a tight connection with the basic linear programming relaxation of a CSP.

Subject Classification

ACM Subject Classification
  • Theory of computation → Constraint and logic programming
  • Constraint Satisfaction Problems
  • Distributed Algorithms
  • Polymorphisms


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