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The Fine-Grained Complexity of Graph Homomorphism Parameterized by Clique-Width

Authors Robert Ganian , Thekla Hamm , Viktoriia Korchemna, Karolina Okrasa , Kirill Simonov

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Author Details

Robert Ganian
  • Algorithms and Complexity Group, TU Wien, Austria
Thekla Hamm
  • Algorithms and Complexity Group, TU Wien, Austria
Viktoriia Korchemna
  • Algorithms and Complexity Group, TU Wien, Austria
Karolina Okrasa
  • Faculty of Matematics and Information Science, Warsaw University of Technology, Poland
  • Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
Kirill Simonov
  • Algorithms and Complexity Group, TU Wien, Austria

Funding

  • Ganian, Robert: Robert Ganian acknowledges support by the Austrian Science Fund (FWF, projects Y1329 and P31336).
  • Hamm, Thekla: Thekla Hamm acknowledges support by the Austrian Science Fund (FWF, projects P31336 and Y1329).
  • Korchemna, Viktoriia: Viktoriia Korchemna acknowledges support by the Austrian Science Fund (FWF, project Y1329).
  • Okrasa, Karolina: Karolina Okrasa acknowledges support by the European Research Council, grant agreement No 714704. Parts of this work were performed while visiting TU Wien, Vienna, Austria.
  • Simonov, Kirill: Kirill Simonov acknowledges support by the Austrian Science Fund (FWF, project P31336).

Cite As Get BibTex

Robert Ganian, Thekla Hamm, Viktoriia Korchemna, Karolina Okrasa, and Kirill Simonov. The Fine-Grained Complexity of Graph Homomorphism Parameterized by Clique-Width. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 66:1-66:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ICALP.2022.66

Abstract

The generic homomorphism problem, which asks whether an input graph G admits a homomorphism into a fixed target graph H, has been widely studied in the literature. In this article, we provide a fine-grained complexity classification of the running time of the homomorphism problem with respect to the clique-width of G (denoted cw) for virtually all choices of H under the Strong Exponential Time Hypothesis. In particular, we identify a property of H called the signature number s(H) and show that for each H, the homomorphism problem can be solved in time O^*(s(H)^cw). Crucially, we then show that this algorithm can be used to obtain essentially tight upper bounds. Specifically, we provide a reduction that yields matching lower bounds for each H that is either a projective core or a graph admitting a factorization with additional properties - allowing us to cover all possible target graphs under long-standing conjectures.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • homomorphism
  • clique-width
  • fine-grained complexity

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