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A Dichotomy for Succinct Representations of Homomorphisms

Authors Christoph Berkholz , Harry Vinall-Smeeth

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Author Details

Christoph Berkholz
  • Technische Universität Ilmenau, Germany
Harry Vinall-Smeeth
  • Humboldt-Universität zu Berlin, Germany

Funding

  • Berkholz, Christoph: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - project number 414325841.
  • Vinall-Smeeth, Harry: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - project number 385256563.

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Christoph Berkholz and Harry Vinall-Smeeth. A Dichotomy for Succinct Representations of Homomorphisms. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 113:1-113:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ICALP.2023.113

Abstract

The task of computing homomorphisms between two finite relational structures A and B is a well-studied question with numerous applications. Since the set Hom(A, B) of all homomorphisms may be very large having a method of representing it in a succinct way, especially one which enables us to perform efficient enumeration and counting, could be extremely useful. One simple yet powerful way of doing so is to decompose Hom(A, B) using union and Cartesian product. Such data structures, called d-representations, have been introduced by Olteanu and Závodný [Olteanu and Závodný, 2015] in the context of database theory. Their results also imply that if the treewidth of the left-hand side structure A is bounded, then a d-representation of polynomial size can be found in polynomial time. We show that for structures of bounded arity this is optimal: if the treewidth is unbounded then there are instances where the size of any d-representation is superpolynomial. Along the way we develop tools for proving lower bounds on the size of d-representations, in particular we define a notion of reduction suitable for this context and prove an almost tight lower bound on the size of d-representations of all k-cliques in a graph.

Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures and algorithms for data management
  • Theory of computation → Complexity theory and logic
  • Theory of computation → Data compression
Keywords
  • homomorphism problem
  • CSP
  • succinct representations
  • enumeration
  • lower bound
  • treewidth

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