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Parameterised Holant Problems

Authors Panagiotis Aivasiliotis, Andreas Göbel , Marc Roth , Johannes Schmitt

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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Mathematics of computing → Graph algorithms
Keywords
  • holant problems
  • counting problems
  • parameterised algorithms
  • fine-grained complexity theory
  • homomorphisms

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Abstract

We investigate the complexity of parameterised holant problems p-Holant(𝒮) for families of symmetric signatures 𝒮. The parameterised holant framework has been introduced by Curticapean in 2015 as a counter-part to the classical and well-established theory of holographic reductions and algorithms, and it constitutes an extensive family of coloured and weighted counting constraint satisfaction problems on graph-like structures, encoding as special cases various well-studied counting problems in parameterised and fine-grained complexity theory such as counting edge-colourful k-matchings, graph-factors, Eulerian orientations or, more generally, subgraphs with weighted degree constraints. We establish an exhaustive complexity trichotomy along the set of signatures 𝒮: Depending on the signatures, p-Holant(𝒮) is either 
1) solvable in "FPT-near-linear time", i.e., in time f(k)⋅ 𝒪̃(|x|), or 
2) solvable in "FPT-matrix-multiplication time", i.e., in time f(k)⋅ {𝒪}(n^{ω}), where n is the number of vertices of the underlying graph, but not solvable in FPT-near-linear time, unless the Triangle Conjecture fails, or 
3) #W[1]-complete and no significant improvement over the naive brute force algorithm is possible unless the Exponential Time Hypothesis fails.  This classification reveals a significant and surprising gap in the complexity landscape of parameterised Holants: Not only is every instance either fixed-parameter tractable or #W[1]-complete, but additionally, every FPT instance is solvable in time (at most) f(k)⋅ {𝒪}(n^{ω}). We show that there are infinitely many instances of each of the types; for example, all constant signatures yield holant problems of type (1), and the problem of counting edge-colourful k-matchings modulo p is of type (p) for p ∈ {2,3}. 
Finally, we also establish a complete classification for a natural uncoloured version of parameterised holant problem p-UnColHolant(𝒮), which encodes as special cases the non-coloured analogues of the aforementioned examples. We show that the complexity of p-UnColHolant(𝒮) is different: Depending on 𝒮 all instances are either solvable in FPT-near-linear time, or #W[1]-complete, that is, there are no instances of type (2).

Cite As Get BibTex

Panagiotis Aivasiliotis, Andreas Göbel, Marc Roth, and Johannes Schmitt. Parameterised Holant Problems. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 7:1-7:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.7

Author Details

Panagiotis Aivasiliotis
  • Hasso Plattner Institute, University of Potsdam, Germany
Andreas Göbel
  • Hasso Plattner Institute, University of Potsdam, Germany
Marc Roth
  • School of Electronic Engineering and Computer Science, Queen Mary University of London, UK
Johannes Schmitt
  • Department of Mathematics, ETH Zürich, Switzerland

Funding

  • Göbel, Andreas: DFG project PAGES (project No. 467516565).
  • Schmitt, Johannes: Swiss National Science Foundation (project No. 219369) and SwissMAP.

Acknowledgements

The authors thank Miriam Backens for their insightful feedback on this work.

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