21 Search Results for "Seppelt, Tim"

Document
DOI: 10.4230/LIPIcs.STACS.2026.25
Homomorphism Indistinguishability, Multiplicity Automata Equivalence, and Polynomial Identity Testing

Authors: Marek Černý and Tim Seppelt

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
Two graphs G and H are homomorphism indistinguishable over a graph class ℱ if they admit the same number of homomorphisms from every graph F ∈ ℱ. Many graph isomorphism relaxations such as (quantum) isomorphism and cospectrality can be characterised as homomorphism indistinguishability over specific graph classes. Thereby, the problems HomInd(ℱ) of deciding homomorphism indistinguishability over ℱ subsume diverse graph isomorphism relaxations whose complexities range from logspace to undecidable. Establishing the first general result on the complexity of HomInd(ℱ), Seppelt (MFCS 2024) showed that HomInd(ℱ) is in randomised polynomial time for every graph class ℱ of bounded treewidth that can be defined in counting monadic second-order logic CMSO₂. We show that this algorithm is conditionally optimal, i.e. it cannot be derandomised unless polynomial identity testing is in P. For CMSO₂-definable graph classes ℱ of bounded pathwidth, we improve the previous complexity upper bound for HomInd(ℱ) from P to C_ = L and show that this is tight. Secondarily, we establish a connection between homomorphism indistinguishability and multiplicity automata equivalence which allows us to pinpoint the complexity of the latter problem as C_ = L-complete.
Cite as

Marek Černý and Tim Seppelt. Homomorphism Indistinguishability, Multiplicity Automata Equivalence, and Polynomial Identity Testing. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 25:1-25:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cerny_et_al:LIPIcs.STACS.2026.25,
  author =	{\v{C}ern\'{y}, Marek and Seppelt, Tim},
  title =	{{Homomorphism Indistinguishability, Multiplicity Automata Equivalence, and Polynomial Identity Testing}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{25:1--25:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.25},
  URN =		{urn:nbn:de:0030-drops-255144},
  doi =		{10.4230/LIPIcs.STACS.2026.25},
  annote =	{Keywords: treewidth, Courcelle’s theorem, logspace, multiplicity automata, polynomial identity testing}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.46
Symmetric Algebraic Circuits and Homomorphism Polynomials

Authors: Anuj Dawar, Benedikt Pago, and Tim Seppelt

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
The central open question of algebraic complexity is whether VP ≠ VNP, which is saying that the permanent cannot be represented by families of polynomial-size algebraic circuits. For symmetric algebraic circuits, this has been confirmed by Dawar and Wilsenach (2020), who showed exponential lower bounds on the size of symmetric circuits for the permanent. In this work, we set out to develop a more general symmetric algebraic complexity theory. Our main result is that a family of symmetric polynomials admits small symmetric circuits if and only if they can be written as a linear combination of homomorphism counting polynomials of graphs of bounded treewidth. We also establish a relationship between the symmetric complexity of subgraph counting polynomials and the vertex cover number of the pattern graph. As a concrete example, we examine the symmetric complexity of immanant families (a generalisation of the determinant and permanent) and show that a known conditional dichotomy due to Curticapean (2021) holds unconditionally in the symmetric setting.
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Anuj Dawar, Benedikt Pago, and Tim Seppelt. Symmetric Algebraic Circuits and Homomorphism Polynomials. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 46:1-46:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{dawar_et_al:LIPIcs.ITCS.2026.46,
  author =	{Dawar, Anuj and Pago, Benedikt and Seppelt, Tim},
  title =	{{Symmetric Algebraic Circuits and Homomorphism Polynomials}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{46:1--46:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.46},
  URN =		{urn:nbn:de:0030-drops-253330},
  doi =		{10.4230/LIPIcs.ITCS.2026.46},
  annote =	{Keywords: algebraic complexity, finite model theory, symmetric circuits, homomorphism counting, graph homomorphism, treewidth, counting width, first-order logic with counting quantifiers}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.32
Vanishing Signatures, Orbit Closure, and the Converse of the Holant Theorem

Authors: Jin-Yi Cai and Ben Young

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
Valiant’s Holant theorem is a powerful tool for algorithms and reductions for counting problems. It states that if two sets ℱ and 𝒢 of tensors (a.k.a. constraint functions or signatures) are related by a holographic transformation, then ℱ and 𝒢 are Holant-indistinguishable, i.e., every tensor network using tensors from ℱ, respectively from 𝒢, contracts to the same value. Xia (ICALP 2010) conjectured the converse of the Holant theorem, but a counterexample was found based on vanishing signatures, those which are Holant-indistinguishable from 0. We prove two near-converses of the Holant theorem using techniques from invariant theory. (I) Holant-indistinguishable ℱ and 𝒢 always admit two sequences of holographic transformations mapping them arbitrarily close to each other, i.e., their GL_q-orbit closures intersect. (II) We show that vanishing signatures are the only true obstacle to a converse of the Holant theorem. As corollaries of the two theorems we obtain the first characterization of homomorphism-indistinguishability over graphs of bounded degree, a long standing open problem, and show that two graphs with invertible adjacency matrices are isomorphic if and only if they are homomorphism-indistinguishable over graphs with maximum degree at most three. We also show that Holant-indistinguishability is complete for a complexity class TOCI introduced by Lysikov and Walter [Vladimir Lysikov and Michael Walter, 2024], and hence hard for graph isomorphism.
Cite as

Jin-Yi Cai and Ben Young. Vanishing Signatures, Orbit Closure, and the Converse of the Holant Theorem. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 32:1-32:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cai_et_al:LIPIcs.ITCS.2026.32,
  author =	{Cai, Jin-Yi and Young, Ben},
  title =	{{Vanishing Signatures, Orbit Closure, and the Converse of the Holant Theorem}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{32:1--32:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.32},
  URN =		{urn:nbn:de:0030-drops-253198},
  doi =		{10.4230/LIPIcs.ITCS.2026.32},
  annote =	{Keywords: Holant, Orbit Closure Intersection, Homomorphism Indistinguishability, Tensor Network}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.61
Quantum Relaxations of CSP and Structure Isomorphism

Authors: Amin Karamlou

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
We investigate quantum relaxations of two key decision problems in computer science: the constraint satisfaction problem (CSP) and the structure isomorphism problem. CSP asks whether a homomorphism exists between two relational structures, while structure isomorphism seeks an isomorphism between them. In recent years, it has become increasingly apparent that many special cases of CSP can be reformulated in terms of the existence of perfect classical strategies in non-local games, a key topic of study in quantum information theory. These games have allowed us to study quantum advantage in relation to many important decision problems, such as the k-colouring problem, and the problem of solving binary constraint systems. Abramsky et al. (2017) have shown that all of these games can be seen as special instances of a non-local CSP game. Moreover, they show that perfect quantum strategies in this CSP game can be viewed as Kleisli morphisms of a graded monad on the category of relational structures, which they dub the quantum monad. In this way, the quantum monad provides a categorical characterisation of quantum advantage for the non-local CSP game. In this work we solidify and expand the results of Abramsky et al., answering several of their open questions. Firstly, we compare the definition of quantum graph homomorphisms arising from this work with an earlier definition of the concept due to Mančinska and Roberson and show that there are graphs which exhibit quantum advantage under one definition but not the other. Our second contribution is to extend the results of Abramsky et al. which only hold in the tensor product framework of quantum mechanics to the commuting operator framework. Next, we study a non-local structure isomorphism game, which generalises the well-studied graph isomorphism game. We show how the construction of the quantum monad can be refined to provide categorical semantics for quantum strategies in this game. This results in a category where morphisms coincide with quantum homomorphisms and isomorphisms coincide with quantum isomorphisms.
Cite as

Amin Karamlou. Quantum Relaxations of CSP and Structure Isomorphism. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 61:1-61:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{karamlou:LIPIcs.MFCS.2025.61,
  author =	{Karamlou, Amin},
  title =	{{Quantum Relaxations of CSP and Structure Isomorphism}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{61:1--61:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.61},
  URN =		{urn:nbn:de:0030-drops-241686},
  doi =		{10.4230/LIPIcs.MFCS.2025.61},
  annote =	{Keywords: CSP, graph isomorphism, quantum information, non-local game, quantum graph homomorphism, monad}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.88
Color Refinement for Relational Structures

Authors: Benjamin Scheidt and Nicole Schweikardt

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
Color Refinement, also known as Naive Vertex Classification, is a classical method to distinguish graphs by iteratively computing a coloring of their vertices. While it is traditionally used as an imperfect way to test for isomorphism, the algorithm has permeated many other, seemingly unrelated, areas of computer science. The method is algorithmically simple, and it has a well-understood distinguishing power: it has been logically characterized by Immerman and Lander (1990) and Cai, Fürer, Immerman (1992), who showed that it distinguishes precisely those graphs that can be distinguished by a sentence of first-order logic with counting quantifiers and only two variables. A combinatorial characterization was given by Dvořák (2010), who showed that it distinguishes precisely those graphs that differ in the number of homomorphisms from some tree. In this paper, we introduce Relational Color Refinement (RCR, for short), a generalization of the Color Refinement method from graphs to arbitrary relational structures, whose distinguishing power admits the equivalent combinatorial and logical characterizations as Color Refinement has on graphs: we show that RCR distinguishes precisely those structures that differ in the number of homomorphisms from an acyclic connected relational structure. Further, we show that RCR distinguishes precisely those structures that are distinguished by a sentence of the guarded fragment of first-order logic with counting quantifiers. Additionally, we show that for every fixed finite relational signature, RCR can be implemented to run on structures of that signature in time O(N⋅log N), where N denotes the number of tuples present in the structure.
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Benjamin Scheidt and Nicole Schweikardt. Color Refinement for Relational Structures. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 88:1-88:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{scheidt_et_al:LIPIcs.MFCS.2025.88,
  author =	{Scheidt, Benjamin and Schweikardt, Nicole},
  title =	{{Color Refinement for Relational Structures}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{88:1--88:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.88},
  URN =		{urn:nbn:de:0030-drops-241958},
  doi =		{10.4230/LIPIcs.MFCS.2025.88},
  annote =	{Keywords: color refinement, counting logics, homomorphism counts, homomorphism indistinguishability, guarded logics, pebble games, relational structures, alpha-acyclicity, join-trees}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.89
Homomorphism Indistinguishability and Game Comonads for Restricted Conjunction and Requantification

Authors: Georg Schindling

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
The notion of homomorphism indistinguishability offers a combinatorial framework for characterizing equivalence relations of graphs, in particular equivalences in counting logics within finite model theory. That is, for certain graph classes, two structures agree on all homomorphism counts from the class if and only if they satisfy the same sentences in a corresponding logic. This perspective often reveals connections between the combinatorial properties of graph classes and the syntactic structure of logical fragments. In this work, we extend this perspective to logics with restricted requantification, refining the stratification of logical resources in finite-variable counting logics. Specifically, we generalize Lovász-type theorems for these logics with either restricted conjunction or bounded quantifier-rank and present new combinatorial proofs of existing results. To this end, we introduce novel path and tree decompositions that incorporate the concept of reusability and develop characterizations based on pursuit-evasion games. Leveraging this framework, we establish that classes of bounded pathwidth and treewidth with reusability constraints are homomorphism distinguishing closed. Finally, we develop a comonadic perspective on requantification by constructing new comonads that encapsulate restricted-reusability pebble games. We show a tight correspondence between their coalgebras and path/tree decompositions, yielding categorical characterizations of reusability in graph decompositions. This unifies logical, combinatorial, and categorical perspectives on the notion of reusability.
Cite as

Georg Schindling. Homomorphism Indistinguishability and Game Comonads for Restricted Conjunction and Requantification. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 89:1-89:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{schindling:LIPIcs.MFCS.2025.89,
  author =	{Schindling, Georg},
  title =	{{Homomorphism Indistinguishability and Game Comonads for Restricted Conjunction and Requantification}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{89:1--89:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.89},
  URN =		{urn:nbn:de:0030-drops-241962},
  doi =		{10.4230/LIPIcs.MFCS.2025.89},
  annote =	{Keywords: homomorphism indistinguishability, game comonads, finite variable counting logic, restricted conjunction, restricted requantification, tree decomposition, path decomposition}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.34
Adaptive Query Algorithms for Relational Structures Based on Homomorphism Counts

Authors: Balder ten Cate, Phokion G. Kolaitis, and Arnar Á. Kristjánsson

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
A query algorithm based on homomorphism counts is a procedure to decide membership for a class of finite relational structures using only homomorphism count queries. A left query algorithm can ask the number of homomorphisms from any structure to the input structure and a right query algorithm can ask the number of homomorphisms from the input structure to any other structure. We systematically compare the expressive power of different types of left or right query algorithms, including non-adaptive query algorithms, adaptive query algorithms that can ask a bounded number of queries, and adaptive query algorithms that can ask an unbounded number of queries. We also consider query algorithms where the homomorphism counting is done over the Boolean semiring 𝔹, meaning that only the existence of a homomorphism is recorded, not the precise number of them.
Cite as

Balder ten Cate, Phokion G. Kolaitis, and Arnar Á. Kristjánsson. Adaptive Query Algorithms for Relational Structures Based on Homomorphism Counts. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 34:1-34:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{tencate_et_al:LIPIcs.MFCS.2025.34,
  author =	{ten Cate, Balder and Kolaitis, Phokion G. and Kristj\'{a}nsson, Arnar \'{A}.},
  title =	{{Adaptive Query Algorithms for Relational Structures Based on Homomorphism Counts}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{34:1--34:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.34},
  URN =		{urn:nbn:de:0030-drops-241413},
  doi =		{10.4230/LIPIcs.MFCS.2025.34},
  annote =	{Keywords: Query algorithms, homomorphisms, homomorphism counts, directed graphs, relational structures, Datalog, constraint satisfaction}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.40
Symmetric Proofs in the Ideal Proof System

Authors: Anuj Dawar, Erich Grädel, Leon Kullmann, and Benedikt Pago

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
We consider the Ideal Proof System (IPS) introduced by Grochow and Pitassi and pose the question of which tautologies admit symmetric proofs, and of what complexity. The symmetry requirement in proofs is inspired by recent work establishing lower bounds in other symmetric models of computation. We link the existence of symmetric IPS proofs to the expressive power of logics such as fixed-point logic with counting and Choiceless Polynomial Time, specifically regarding the graph isomorphism problem. We identify relationships and tradeoffs between the symmetry of proofs and other parameters of IPS proofs such as size, degree and linearity. We study these on a number of standard families of tautologies from proof complexity and finite model theory such as the pigeonhole principle, the subset sum problem and the Cai-Fürer-Immerman graphs, exhibiting non-trivial upper bounds on the size of symmetric IPS proofs.
Cite as

Anuj Dawar, Erich Grädel, Leon Kullmann, and Benedikt Pago. Symmetric Proofs in the Ideal Proof System. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 40:1-40:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dawar_et_al:LIPIcs.MFCS.2025.40,
  author =	{Dawar, Anuj and Gr\"{a}del, Erich and Kullmann, Leon and Pago, Benedikt},
  title =	{{Symmetric Proofs in the Ideal Proof System}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{40:1--40:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.40},
  URN =		{urn:nbn:de:0030-drops-241477},
  doi =		{10.4230/LIPIcs.MFCS.2025.40},
  annote =	{Keywords: proof complexity, algebraic complexity, descriptive complexity, symmetric circuits, graph isomorphism}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.105
NPA Hierarchy for Quantum Isomorphism and Homomorphism Indistinguishability

Authors: Prem Nigam Kar, David E. Roberson, Tim Seppelt, and Peter Zeman

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

Abstract
Mančinska and Roberson [FOCS'20] showed that two graphs are quantum isomorphic if and only if they are homomorphism indistinguishable over the class of planar graphs. Atserias et al. [JCTB'19] proved that quantum isomorphism is undecidable in general. The NPA hierarchy gives a sequence of semidefinite programming relaxations of quantum isomorphism. Recently, Roberson and Seppelt [ICALP'23] obtained a homomorphism indistinguishability characterization of the feasibility of each level of the Lasserre hierarchy of semidefinite programming relaxations of graph isomorphism. We prove a quantum analogue of this result by showing that each level of the NPA hierarchy of SDP relaxations for quantum isomorphism of graphs is equivalent to homomorphism indistinguishability over an appropriate class of planar graphs. By combining the convergence of the NPA hierarchy with the fact that the union of these graph classes is the set of all planar graphs, we are able to give a new proof of the result of Mančinska and Roberson [FOCS'20] that avoids the use of the theory of quantum groups. This homomorphism indistinguishability characterization also allows us to give a randomized polynomial-time algorithm deciding exact feasibility of each fixed level of the NPA hierarchy of SDP relaxations for quantum isomorphism.
Cite as

Prem Nigam Kar, David E. Roberson, Tim Seppelt, and Peter Zeman. NPA Hierarchy for Quantum Isomorphism and Homomorphism Indistinguishability. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 105:1-105:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kar_et_al:LIPIcs.ICALP.2025.105,
  author =	{Kar, Prem Nigam and Roberson, David E. and Seppelt, Tim and Zeman, Peter},
  title =	{{NPA Hierarchy for Quantum Isomorphism and Homomorphism Indistinguishability}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{105:1--105:19},
  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.2025.105},
  URN =		{urn:nbn:de:0030-drops-234828},
  doi =		{10.4230/LIPIcs.ICALP.2025.105},
  annote =	{Keywords: Quantum isomorphism, NPA hierarchy, homomorphism indistinguishability}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.136
The Converse of the Real Orthogonal Holant Theorem

Authors: Ben Young

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

Abstract
The Holant theorem is a powerful tool for studying the computational complexity of counting problems. Due to the great expressiveness of the Holant framework, a converse to the Holant theorem would itself be a very powerful counting indistinguishability theorem. The most general converse does not hold, but we prove the following, still highly general, version: if any two sets of real-valued signatures are Holant-indistinguishable, then they are equivalent up to an orthogonal transformation. This resolves a partially open conjecture of Xia (2010). Consequences of this theorem include the well-known result that homomorphism counts from all graphs determine a graph up to isomorphism, the classical sufficient condition for simultaneous orthogonal similarity of sets of real matrices, and a combinatorial characterization of sets of simultaneosly orthogonally decomposable (odeco) symmetric tensors.
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Ben Young. The Converse of the Real Orthogonal Holant Theorem. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 136:1-136:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{young:LIPIcs.ICALP.2025.136,
  author =	{Young, Ben},
  title =	{{The Converse of the Real Orthogonal Holant Theorem}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{136:1--136:20},
  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.2025.136},
  URN =		{urn:nbn:de:0030-drops-235138},
  doi =		{10.4230/LIPIcs.ICALP.2025.136},
  annote =	{Keywords: Holant, Counting Indistinguishability, Odeco}
}
Document
DOI: 10.4230/LIPIcs.CSL.2025.6
Equi-Rank Homomorphism Preservation Theorem on Finite Structures

Authors: Benjamin Rossman

Published in: LIPIcs, Volume 326, 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)

Abstract
The Homomorphism Preservation Theorem (HPT) of classical model theory states that a first-order sentence is preserved under homomorphisms if, and only if, it is equivalent to an existential-positive sentence. This theorem remains valid when restricted to finite structures, as demonstrated by the author in [Rossman, 2008; Rossman, 2017] via distinct model-theoretic and circuit-complexity based proofs. In this paper, we present a third (and significantly simpler) proof of the finitary HPT based on a generalized Cai-Fürer-Immerman construction. This method establishes a tight correspondence between syntactic parameters of a homomorphism-preserved sentence (quantifier rank, variable width, alternation height) and structural parameters of its minimal models (tree-width, tree-depth, decomposition height). Consequently, we prove a conjectured "equi-rank" version of the finitary HPT. In contrast, previous versions of the finitary HPT possess additional properties, but incur blow-ups in the quantifier rank of the equivalent existential-positive sentence.
Cite as

Benjamin Rossman. Equi-Rank Homomorphism Preservation Theorem on Finite Structures. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{rossman:LIPIcs.CSL.2025.6,
  author =	{Rossman, Benjamin},
  title =	{{Equi-Rank Homomorphism Preservation Theorem on Finite Structures}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{6:1--6:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-362-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{326},
  editor =	{Endrullis, J\"{o}rg and Schmitz, Sylvain},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.6},
  URN =		{urn:nbn:de:0030-drops-227634},
  doi =		{10.4230/LIPIcs.CSL.2025.6},
  annote =	{Keywords: finite model theory, preservation theorems, quantifier rank}
}
Document
DOI: 10.4230/LIPIcs.CSL.2025.13
Computational Complexity of the Weisfeiler-Leman Dimension

Authors: Moritz Lichter, Simon Raßmann, and Pascal Schweitzer

Published in: LIPIcs, Volume 326, 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)

Abstract
The Weisfeiler-Leman dimension of a graph G is the least number k such that the k-dimensional Weisfeiler-Leman algorithm distinguishes G from every other non-isomorphic graph, or equivalently, the least k such that G is definable in (k+1)-variable first-order logic with counting. The dimension is a standard measure of the descriptive or structural complexity of a graph and recently finds various applications in particular in the context of machine learning. This paper studies the complexity of computing the Weisfeiler-Leman dimension. We observe that deciding whether the Weisfeiler-Leman dimension of G is at most k is NP-hard, even if G is restricted to have 4-bounded color classes. For each fixed k ≥ 2, we give a polynomial-time algorithm that decides whether the Weisfeiler-Leman dimension of a given graph with 5-bounded color classes is at most k. Moreover, we show that for these bounds on the color classes, this is optimal because the problem is PTIME-hard under logspace-uniform AC_0-reductions. Furthermore, for each larger bound c on the color classes and each fixed k ≥ 2, we provide a polynomial-time decision algorithm for the abelian case, that is, for structures of which each color class has an abelian automorphism group. While the graph classes we consider may seem quite restrictive, graphs with 4-bounded abelian colors include CFI-graphs and multipedes, which form the basis of almost all known hard instances and lower bounds related to the Weisfeiler-Leman algorithm.
Cite as

Moritz Lichter, Simon Raßmann, and Pascal Schweitzer. Computational Complexity of the Weisfeiler-Leman Dimension. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 13:1-13:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{lichter_et_al:LIPIcs.CSL.2025.13,
  author =	{Lichter, Moritz and Ra{\ss}mann, Simon and Schweitzer, Pascal},
  title =	{{Computational Complexity of the Weisfeiler-Leman Dimension}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{13:1--13:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-362-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{326},
  editor =	{Endrullis, J\"{o}rg and Schmitz, Sylvain},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.13},
  URN =		{urn:nbn:de:0030-drops-227707},
  doi =		{10.4230/LIPIcs.CSL.2025.13},
  annote =	{Keywords: Weisfeiler-Leman algorithm, dimension, complexity, coherent configurations}
}
Document
DOI: 10.4230/LIPIcs.CSL.2025.14
Finite Variable Counting Logics with Restricted Requantification

Authors: Simon Raßmann, Georg Schindling, and Pascal Schweitzer

Published in: LIPIcs, Volume 326, 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)

Abstract
Counting logics with a bounded number of variables form one of the central concepts in descriptive complexity theory. Although they restrict the number of variables that a formula can contain, the variables can be nested within scopes of quantified occurrences of themselves. In other words, the variables can be requantified. We study the fragments obtained from counting logics by restricting requantification for some but not necessarily all the variables. Similar to the logics without limitation on requantification, we develop tools to investigate the restricted variants. Specifically, we introduce a bijective pebble game in which certain pebbles can only be placed once and for all, and a corresponding two-parametric family of Weisfeiler-Leman algorithms. We show close correspondences between the three concepts. By using a suitable cops-and-robber game and adaptations of the Cai-Fürer-Immerman construction, we completely clarify the relative expressive power of the new logics. We show that the restriction of requantification has beneficial algorithmic implications in terms of graph identification. Indeed, we argue that with regard to space complexity, non-requantifiable variables only incur an additive polynomial factor when testing for equivalence. In contrast, for all we know, requantifiable variables incur a multiplicative linear factor. Finally, we observe that graphs of bounded tree-depth and 3-connected planar graphs can be identified using no, respectively, only a very limited number of requantifiable variables.
Cite as

Simon Raßmann, Georg Schindling, and Pascal Schweitzer. Finite Variable Counting Logics with Restricted Requantification. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 14:1-14:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{ramann_et_al:LIPIcs.CSL.2025.14,
  author =	{Ra{\ss}mann, Simon and Schindling, Georg and Schweitzer, Pascal},
  title =	{{Finite Variable Counting Logics with Restricted Requantification}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{14:1--14:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-362-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{326},
  editor =	{Endrullis, J\"{o}rg and Schmitz, Sylvain},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.14},
  URN =		{urn:nbn:de:0030-drops-227716},
  doi =		{10.4230/LIPIcs.CSL.2025.14},
  annote =	{Keywords: Requantification, Finite variable counting logics, Weisfeiler-Leman algorithm}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.82
An Algorithmic Meta Theorem for Homomorphism Indistinguishability

Authors: Tim Seppelt

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
Two graphs G and H are homomorphism indistinguishable over a family of graphs ℱ if for all graphs F ∈ ℱ the number of homomorphisms from F to G is equal to the number of homomorphism from F to H. Many natural equivalence relations comparing graphs such as (quantum) isomorphism, cospectrality, and logical equivalences can be characterised as homomorphism indistinguishability relations over various graph classes. The wealth of such results motivates a more fundamental study of homomorphism indistinguishability. From a computational perspective, the central object of interest is the decision problem HomInd(ℱ) which asks to determine whether two input graphs G and H are homomorphism indistinguishable over a fixed graph class ℱ. The problem HomInd(ℱ) is known to be decidable only for few graph classes ℱ. Due to a conjecture by Roberson (2022) and results by Seppelt (MFCS 2023), homomorphism indistinguishability relations over minor-closed graph classes are of special interest. We show that HomInd(ℱ) admits a randomised polynomial-time algorithm for every minor-closed graph class ℱ of bounded treewidth. This result extends to a version of HomInd where the graph class ℱ is specified by a sentence in counting monadic second-order logic and a bound k on the treewidth, which are given as input. For fixed k, this problem is randomised fixed-parameter tractable. If k is part of the input, then it is coNP- and coW[1]-hard. Addressing a problem posed by Berkholz (2012), we show coNP-hardness by establishing that deciding indistinguishability under the k-dimensional Weisfeiler-Leman algorithm is coNP-hard when k is part of the input.
Cite as

Tim Seppelt. An Algorithmic Meta Theorem for Homomorphism Indistinguishability. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 82:1-82:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{seppelt:LIPIcs.MFCS.2024.82,
  author =	{Seppelt, Tim},
  title =	{{An Algorithmic Meta Theorem for Homomorphism Indistinguishability}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{82:1--82:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.82},
  URN =		{urn:nbn:de:0030-drops-206387},
  doi =		{10.4230/LIPIcs.MFCS.2024.82},
  annote =	{Keywords: homomorphism indistinguishability, graph homomorphism, graph minor, recognisability, randomised algorithm, Courcelle’s Theorem}
}
Document
DOI: 10.4230/LIPIcs.STACS.2024.19
The Complexity of Homomorphism Reconstructibility

Authors: Jan Böker, Louis Härtel, Nina Runde, Tim Seppelt, and Christoph Standke

Published in: LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

Abstract
Representing graphs by their homomorphism counts has led to the beautiful theory of homomorphism indistinguishability in recent years. Moreover, homomorphism counts have promising applications in database theory and machine learning, where one would like to answer queries or classify graphs solely based on the representation of a graph G as a finite vector of homomorphism counts from some fixed finite set of graphs to G. We study the computational complexity of the arguably most fundamental computational problem associated to these representations, the homomorphism reconstructability problem: given a finite sequence of graphs and a corresponding vector of natural numbers, decide whether there exists a graph G that realises the given vector as the homomorphism counts from the given graphs. We show that this problem yields a natural example of an NP^#𝖯-hard problem, which still can be NP-hard when restricted to a fixed number of input graphs of bounded treewidth and a fixed input vector of natural numbers, or alternatively, when restricted to a finite input set of graphs. We further show that, when restricted to a finite input set of graphs and given an upper bound on the order of the graph G as additional input, the problem cannot be NP-hard unless 𝖯 = NP. For this regime, we obtain partial positive results. We also investigate the problem’s parameterised complexity and provide fpt-algorithms for the case that a single graph is given and that multiple graphs of the same order with subgraph instead of homomorphism counts are given.
Cite as

Jan Böker, Louis Härtel, Nina Runde, Tim Seppelt, and Christoph Standke. The Complexity of Homomorphism Reconstructibility. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 19:1-19:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{boker_et_al:LIPIcs.STACS.2024.19,
  author =	{B\"{o}ker, Jan and H\"{a}rtel, Louis and Runde, Nina and Seppelt, Tim and Standke, Christoph},
  title =	{{The Complexity of Homomorphism Reconstructibility}},
  booktitle =	{41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)},
  pages =	{19:1--19:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-311-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{289},
  editor =	{Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.19},
  URN =		{urn:nbn:de:0030-drops-197298},
  doi =		{10.4230/LIPIcs.STACS.2024.19},
  annote =	{Keywords: graph homomorphism, counting complexity, parameterised complexity}
}
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