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DOI: 10.4230/LIPIcs.ICALP.2025.148

Holant* Dichotomy on Domain Size 3: A Geometric Perspective

Authors: Jin-Yi Cai and Jin Soo Ihm

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract

Holant problems are a general framework to study the computational complexity of counting problems. It is a more expressive framework than counting constraint satisfaction problems (CSP) which are in turn more expressive than counting graph homomorphisms (GH). In this paper, we prove the first complexity dichotomy of Holant^*₃(ℱ) where ℱ is an arbitrary set of symmetric, real valued constraint functions on domain size 3. We give an explicit tractability criterion and prove that, if ℱ satisfies this criterion then Holant^*₃(ℱ) is polynomial time computable, and otherwise it is #P-hard, with no intermediate cases. We show that the geometry of the tensor decomposition of the constraint functions plays a central role in the formulation as well as the structural internal logic of the dichotomy.

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Jin-Yi Cai and Jin Soo Ihm. Holant* Dichotomy on Domain Size 3: A Geometric Perspective. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 148:1-148:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cai_et_al:LIPIcs.ICALP.2025.148,
  author =	{Cai, Jin-Yi and Ihm, Jin Soo},
  title =	{{Holant* Dichotomy on Domain Size 3: A Geometric Perspective}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{148:1--148:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.148},
  URN =		{urn:nbn:de:0030-drops-235254},
  doi =		{10.4230/LIPIcs.ICALP.2025.148},
  annote =	{Keywords: Holant problem, Complexity dichotomy, Higher domain}
}

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Holant* Dichotomy on Domain Size 3: A Geometric Perspective

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