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LIPIcs, Volume 386
51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)
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Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Mathematical Foundations of Computer Science (MFCS)
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- published at: 2026-08-21
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-442-0
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Complete Volume
LIPIcs, Volume 386, MFCS 2026, Complete Volume
Abstract
LIPIcs, Volume 386, MFCS 2026, Complete Volume
Cite as
51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 1-1660, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@Proceedings{koucky_et_al:LIPIcs.MFCS.2026,
title = {{LIPIcs, Volume 386, MFCS 2026, Complete Volume}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {1--1660},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026},
URN = {urn:nbn:de:0030-drops-276644},
doi = {10.4230/LIPIcs.MFCS.2026},
annote = {Keywords: LIPIcs, Volume 386, MFCS 2026, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 0:i-0:xxx, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{koucky_et_al:LIPIcs.MFCS.2026.0,
author = {Kouck\'{y}, Michal and Petrișan, Daniela},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {0:i--0:xxx},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.0},
URN = {urn:nbn:de:0030-drops-276636},
doi = {10.4230/LIPIcs.MFCS.2026.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Invited Talk
Homotopy Theory in Complexity of the Graph Homomorphism Problem (Invited Talk)
Abstract
I will talk about an emerging application of homotopy theory in computational complexity of combinatorial problems, more precisely homomorphism problems. Homomorphism problems appear under many names, including constraint satisfaction problems and conjunctive database queries.
A prime example of a homomorphism problem is graph colouring; by a colouring of a graph with k colours, I mean an assignment of colours to vertices under which no edge is monochromatic - this is equivalent to the existence of a homomorphism to the clique with k vertices. Graph 3-colouring is a prototypical example of an NP-complete problem. There are many variations on this problem whose complexity remains widely open. For example, although it is generally believed that colouring a 3-colourable graph with a fixed number of colours is NP-hard, only the hardness of colouring of such a graph with 5 colours is known (and shown only in 2019).
I will give an overview of several related results about variations of graph colouring that share a common theme of using a method based on tools from topological combinatorics and on ideas of Lovász [J. Comb. Theory, Ser. A, 25(3):319-324, 1978].
Cite as
Jakub Opršal. Homotopy Theory in Complexity of the Graph Homomorphism Problem (Invited Talk). In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, p. 1:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{oprsal:LIPIcs.MFCS.2026.1,
author = {Opr\v{s}al, Jakub},
title = {{Homotopy Theory in Complexity of the Graph Homomorphism Problem}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {1:1--1:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.1},
URN = {urn:nbn:de:0030-drops-273829},
doi = {10.4230/LIPIcs.MFCS.2026.1},
annote = {Keywords: homomorphism problem, constraint satisfaction problem, graph colouring, topological methods}
}
Document
Invited Talk
Highly-Efficient Local Proofs and Codes (Invited Talk)
Abstract
Interactive oracle proofs (IOPs) extend the classical notion of probabilistically-checkable proofs (PCPs) by allowing a verifier to interact with a prover over a small number of rounds, while querying the prover’s messages in only a few locations.
A recent line of work gave highly-efficient IOPs outperforming state-of-the-art PCPs, for example, constant-round and constant-query (ZK-)IOPs with only a linear (and even approaching the witness length) amount of communication, as well as (ZK-)IOPs with linear-time prover complexity. These constructions were leveraged in turn to obtain highly-efficient succinct arguments and zero-knowledge proofs.
The improved efficiency was obtained by replacing polynomial-based codes, commonly used in such proof systems, with more efficient (tensor-based) codes. In particular, these constructions bypassed a barrier imposed by the need to encode the computation using a multiplication code.
In the talk I will survey these highly-efficient IOP constructions, and highlight some interesting open problems raised by these works.
Cite as
Noga Ron-Zewi. Highly-Efficient Local Proofs and Codes (Invited Talk). In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, p. 2:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ronzewi:LIPIcs.MFCS.2026.2,
author = {Ron-Zewi, Noga},
title = {{Highly-Efficient Local Proofs and Codes}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {2:1--2:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.2},
URN = {urn:nbn:de:0030-drops-273836},
doi = {10.4230/LIPIcs.MFCS.2026.2},
annote = {Keywords: Interactive oracle proofs, probabilistically-checkable proofs, error-correcting codes}
}
Document
Invited Talk
Some Recent Developments in Space Complexity (Invited Talk)
Abstract
Given a function, what is the minimal memory necessary to compute it? We will describe some old [John E. Hopcroft et al., 1977; Wolfgang J. Paul and Rüdiger Reischuk, 1981; Joseph Y. Halpern et al., 1986] and new algorithmic developments that give surprisingly low-space solutions to this question in many cases. We will survey the recent proof [Ryan Williams, 2026] that TIME[t] is contained in SPACE[√{t log t}] for the multitape Turing machine model, and the engine that makes the proof possible: the amazing Cook-Mertz algorithm [James Cook and Ian Mertz, 2024] for a problem called Tree Evaluation [Stephen A. Cook et al., 2012]. We will also briefly outline some new developments and generalizations that we've recently proved, pushing the low-space frontier beyond multitape Turing machines. The latter is joint work with Danil Sibgatullin (to appear).
Cite as
R. Ryan Williams. Some Recent Developments in Space Complexity (Invited Talk). In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, p. 3:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{williams:LIPIcs.MFCS.2026.3,
author = {Williams, R. Ryan},
title = {{Some Recent Developments in Space Complexity}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {3:1--3:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.3},
URN = {urn:nbn:de:0030-drops-273845},
doi = {10.4230/LIPIcs.MFCS.2026.3},
annote = {Keywords: time lower bound, space complexity, multitape Turing machine, P versus PSPACE, tree evaluation problem}
}
Document
Invited Talk
The Art of Balance: Many Facets of Dyck Recognition (Invited Talk)
Abstract
The Dyck language, consisting of well-balanced parenthesis sequences, is one of the central objects in formal language theory. The Dyck languages appear naturally in numerous applications: balanced-parenthesis encodings succinctly represent rooted trees, programming languages rely heavily on nested structures, and structured data formats such as XML often utilize a notion of balanced parenthesis sequences.
Dyck languages also arise in computational biology. RNA and DNA secondary structures can often be viewed as "almost balanced" sequences, so understanding the behaviour of the Dyck languages is often an important building block for designing algorithms on such sequences.
In this talk, I will survey several recent developments in Dyck language recognition, highlighting surprising connections to different areas of TCS, including regular language recognition, Boolean matrix multiplication, pattern matching, and graph algorithms. I will also discuss some of the major open questions in the area.
Cite as
Tatiana Starikovskaya. The Art of Balance: Many Facets of Dyck Recognition (Invited Talk). In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, p. 4:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{starikovskaya:LIPIcs.MFCS.2026.4,
author = {Starikovskaya, Tatiana},
title = {{The Art of Balance: Many Facets of Dyck Recognition}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {4:1--4:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.4},
URN = {urn:nbn:de:0030-drops-273852},
doi = {10.4230/LIPIcs.MFCS.2026.4},
annote = {Keywords: Formal language recognition, Dyck languages, Boolean matrix multiplication, pattern matching, graph algorithms}
}
Document
Invited Talk
Diagrammatic Reasoning, Formally (Invited Talk)
Abstract
Modern proof assistants make it possible to verify theorems, but often makes it harder than with pen and paper. This is especially true in domains where proofs are best depicted using diagrams. For instance, confluence diagrams in rewriting theory, commuting diagrams in category theory, or string diagrams in monoidal categories. By using various tools and techniques to solve well-defined classes of goals, infer appropriate data, or perform high-level reasoning steps [Dexter Kozen, 1997; André Joyal and Ross Street, 1991], I will show how to obtain robust and elegant proof scripts in these application domains [Damien Pous, 2013; Damien Pous, 2026].
Cite as
Damien Pous. Diagrammatic Reasoning, Formally (Invited Talk). In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, p. 5:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{pous:LIPIcs.MFCS.2026.5,
author = {Pous, Damien},
title = {{Diagrammatic Reasoning, Formally}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {5:1--5:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.5},
URN = {urn:nbn:de:0030-drops-273867},
doi = {10.4230/LIPIcs.MFCS.2026.5},
annote = {Keywords: Formal proofs, graphical proofs, string diagrams, monoidal categories, Kleene algebra, relation algebra, Rocq}
}
Document
A Complete Equational Presentation of Qudit Circuits via Polycontrolled PROPs
Abstract
High-dimensional quantum computation needs a native circuit-level equational theory for qudits. We give the first finite schematic equational theory that is sound and complete for exact unitary qudit circuits in every finite dimension at least two. Circuits are built from local gates, sequential and parallel composition, and value-controls; equality is derivable exactly when the standard unitary denotations agree. For each dimension, a finite list of local bounded-arity axiom schemata presents the theory, and the diagrammatic shapes do not depend on d. Primitive value-control makes control on a chosen basis value part of the language, so local rules generate the internal algebra of controlled operations within the circuit PROP. This gives a finite, dimension-uniform basis for exact equational reasoning about qudit circuits.
Cite as
Colin Blake. A Complete Equational Presentation of Qudit Circuits via Polycontrolled PROPs. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{blake:LIPIcs.MFCS.2026.6,
author = {Blake, Colin},
title = {{A Complete Equational Presentation of Qudit Circuits via Polycontrolled PROPs}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {6:1--6:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.6},
URN = {urn:nbn:de:0030-drops-273875},
doi = {10.4230/LIPIcs.MFCS.2026.6},
annote = {Keywords: Qudit circuits, Quantum circuits, Completeness, Control, Categorical quantum mechanics}
}
Document
A Congestion Parameter for Depth-First Graph Traversals
Abstract
We explore a new graph parameter, KLX number, which quantifies the minimum edge congestion of depth-first search (DFS) traversals of a given graph. Originally motivated by a problem in RNA nanostructure design, this parameter is also of independent theoretical interest. Informally, the KLX number of a graph is defined as the minimum, over all its DFS traversals, of the maximum number of back edges that are simultaneously open during the traversal.
We provide full characterisations and linear-time recognition algorithms for graphs with KLX numbers 0, 1 and 2. We also relate KLX to tree-width, proving that any graph satisfies TW ≤ KLX+1. Furthermore, we show that the property KLX ≤ k is MSO₂-expressible for every fixed k. Combined with the tree-width bound, this result implies that determining whether a graph has KLX number at most k can be achieved in linear time for any constant k.
Cite as
Codaline Bourotte, Gwendal Ducloz, Pekka Orponen, and Shinnosuke Seki. A Congestion Parameter for Depth-First Graph Traversals. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bourotte_et_al:LIPIcs.MFCS.2026.7,
author = {Bourotte, Codaline and Ducloz, Gwendal and Orponen, Pekka and Seki, Shinnosuke},
title = {{A Congestion Parameter for Depth-First Graph Traversals}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {7:1--7:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.7},
URN = {urn:nbn:de:0030-drops-273889},
doi = {10.4230/LIPIcs.MFCS.2026.7},
annote = {Keywords: KLX, depth-first search, DFS trees, k-connectedness, tree-width, parameterised complexity, monadic second-order logic, Courcelle’s theorem, RNA nanotechnology}
}
Document
A Dividing Line for Structural Kernelization of Component Order Connectivity via Distance to Bounded Pathwidth
Abstract
In this work we study a classic generalization of the ubiquitous Vertex Cover (VC) problem, called the Component Order Connectivity (COC) problem. In COC, given an undirected graph G, integers d ≥ 1 and k, the goal is to determine if there is a set of at most k vertices whose deletion results in a graph where each connected component has at most d vertices. When d = 1, this is exactly VC.
This work is inspired by polynomial kernelization results with respect to structural parameters for VC. On one hand, Jansen & Bodlaender [TOCS 2013] show that VC admits a polynomial kernel when the parameter is the distance to treewidth-1 graphs, on the other hand Cygan, Lokshtanov, Pilipczuk, Pilipczuk & Saurabh [TOCS 2014] showed that VC does not admit a polynomial kernel when the parameter is distance to treewidth-2 graphs.
Greilhuber & Sharma [IPEC 2024] showed that, for any d ≥ 2, d-COC cannot admit a polynomial kernel when the parameter is distance to a forest of pathwidth 2. Here, d-COC is the variant of COC where d is a fixed constant rather than part of the input. We complement this result and show that, analogously to the VC setting, where distance to treewidth-1 graphs versus distance to treewidth-2 graphs is the dividing line between structural parameterizations that admit and respectively do not admit polynomial kernelization, for COC this dividing line lies between distance to pathwidth-1 graphs and distance to pathwidth-2 graphs. The main technical result of this work is that COC admits a polynomial kernel parameterized by distance to pathwidth-1 graphs plus d.
The problem d-COC can also be expressed as an ℱ-MinorDeletion problem for an appropriate graph family ℱ. One of the central questions around ℱ-MinorDeletion is for which families ℱ and minor-closed graph classes 𝒢 the problem admits a polynomial kernel when parameterized by the distance to 𝒢. For some families ℱ complete dichotomies answering this question are known [Bougeret et al., SIDMA 2022][Bougeret et al., STACS 2026][Bougeret et al., arXiv 2026]. But, these results do not capture the 2-COC problem. We show that, when d ≥ 2, the line of tractability for polynomial kernelization of d-COC parameterized by the distance to 𝒢 is different from the tractability line of the ℱ-MinorDeletion problems for which the currently known dichotomies apply. Thus, with our result, d-COC serves as an outlier in the class of ℱ-MinorDeletion problems when it comes to understanding the dichotomies for polynomial kernelization when parameterizing by the distance to some minor-closed graph class.
Cite as
Jakob Greilhuber and Roohani Sharma. A Dividing Line for Structural Kernelization of Component Order Connectivity via Distance to Bounded Pathwidth. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 8:1-8:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{greilhuber_et_al:LIPIcs.MFCS.2026.8,
author = {Greilhuber, Jakob and Sharma, Roohani},
title = {{A Dividing Line for Structural Kernelization of Component Order Connectivity via Distance to Bounded Pathwidth}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {8:1--8:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.8},
URN = {urn:nbn:de:0030-drops-273890},
doi = {10.4230/LIPIcs.MFCS.2026.8},
annote = {Keywords: Kernelization, Component Order Connectivity, Caterpillars, Pathwidth, Structural Parameterization}
}
Document
A Forward-Only Construction of Semilinear Inductive Invariants for VAS
Abstract
The reachability problem for Vector Addition Systems (VAS) is a central decision problem in the theory of infinite-state systems, first solved by Kosaraju and Mayr in the 1980s. An alternative, conceptually simpler approach introduced by Leroux shows that non-reachability is always witnessed by semilinear inductive invariants, yielding a decision procedure by combining an enumeration of runs with a search for such invariants. However, the construction of these invariants relies on a back-and-forth scheme that depends symmetrically on the source and the target. As a result, the invariants are not guaranteed to reflect the structural properties of the VAS, and the construction is difficult to extend to asymmetric models such as Branching VAS.
We introduce a new forward-only construction of semilinear inductive invariants for VAS. Our method builds invariants from the source configuration alone and avoids the need for backward reasoning. This yields invariants that are more canonical and better aligned with the structure of the system. In particular, our method produces periodic inductive invariants for periodic VAS.
Beyond its intrinsic interest, our approach provides a step toward extending invariant-based techniques to Branching VAS.
Cite as
Clotilde Bizière, Jérôme Leroux, and Grégoire Sutre. A Forward-Only Construction of Semilinear Inductive Invariants for VAS. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 9:1-9:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{biziere_et_al:LIPIcs.MFCS.2026.9,
author = {Bizi\`{e}re, Clotilde and Leroux, J\'{e}r\^{o}me and Sutre, Gr\'{e}goire},
title = {{A Forward-Only Construction of Semilinear Inductive Invariants for VAS}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {9:1--9:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.9},
URN = {urn:nbn:de:0030-drops-273905},
doi = {10.4230/LIPIcs.MFCS.2026.9},
annote = {Keywords: Vector addition systems, Inductive invariants, Semilinear sets, Verification}
}
Document
A Framework for Ruling out Quantum Speedups
Abstract
We study when partial Boolean functions can (and cannot) exhibit superpolynomial quantum query speedups, and develop a general framework for ruling out such speedups via two complementary lenses: promise-aware complexity measures and function completions.
First, we introduce promise versions of standard combinatorial measures (including block sensitivity and related variants) and prove that if the relevant promise and completion measures "collapse", then deterministic and quantum query complexities are necessarily polynomially related, i.e., D(f) = poly(Q(f)). We then analyze structured families of promises, including symmetric partial functions and promises supported on Hamming slices, obtaining sharp (up to polynomial factors) characterizations in terms of a single gap parameter for the symmetric case and refined slice-dependent bounds for k-slice domains.
Next, we formalize completion complexity as the minimum of a measure over total completions of a partial function, and show that completability of a measure captures the possibility of superpolynomial quantum speedups. Finally, we apply this viewpoint to derive broad non-speedup criteria for some classes of functions admitting well-behaved completions, such as functions with low maximum influence on both the standard and p-biased hypercubes and functions with efficiently identifiable domains, and then show some hardness results for general completion techniques.
Cite as
Thomas Huffstutler, Upendra Kapshikar, David Miloschewsky, and Supartha Podder. A Framework for Ruling out Quantum Speedups. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{huffstutler_et_al:LIPIcs.MFCS.2026.10,
author = {Huffstutler, Thomas and Kapshikar, Upendra and Miloschewsky, David and Podder, Supartha},
title = {{A Framework for Ruling out Quantum Speedups}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {10:1--10:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.10},
URN = {urn:nbn:de:0030-drops-273910},
doi = {10.4230/LIPIcs.MFCS.2026.10},
annote = {Keywords: Complexity, Boolean Analysis, Quantum Speedup}
}
Document
A Near-Linear-Time Algorithm for Finding a Well-Spread Perfect Matching in Bridgeless Cubic Graphs
Abstract
We present a near-linear-time algorithm that, given a bridgeless cubic graph, finds a perfect matching intersecting every 3-edge-cut in exactly one edge. This improves over a cubic algorithm of Boyd et al. for the same problem, and over our previous algorithm, which worked only for 3-edge-connected graphs. The main ingredient is a cactus representation of the 2-edge-cuts, together with an efficient update procedure under 2-cut reductions.
Cite as
Babak Ghanbari and Robert Šámal. A Near-Linear-Time Algorithm for Finding a Well-Spread Perfect Matching in Bridgeless Cubic Graphs. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ghanbari_et_al:LIPIcs.MFCS.2026.11,
author = {Ghanbari, Babak and \v{S}\'{a}mal, Robert},
title = {{A Near-Linear-Time Algorithm for Finding a Well-Spread Perfect Matching in Bridgeless Cubic Graphs}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {11:1--11:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.11},
URN = {urn:nbn:de:0030-drops-273922},
doi = {10.4230/LIPIcs.MFCS.2026.11},
annote = {Keywords: bridgeless cubic graphs, well-spread perfect matching, edge cuts, cactus representation}
}
Document
A Slightly Improved Upper Bound for Quantum Statistical Zero-Knowledge
Abstract
The complexity class Quantum Statistical Zero-Knowledge (QSZK), introduced by Watrous (FOCS 2002) and later refined in Watrous (SICOMP, 2009), has the best known upper bound QIP(2) ∩ co-QIP(2), which was simplified following the inclusion QIP(2) ⊆ PSPACE established in Jain, Upadhyay, and Watrous (FOCS 2009). Here, QIP(2) denotes the class of promise problems that admit two-message quantum interactive proof systems in which the honest prover is typically computationally unbounded, and co-QIP(2) denotes the complement of QIP(2).
We slightly improve this upper bound to QIP(2) ∩ co-QIP(2) with a quantum linear-space honest prover. Specifically, the honest prover uses space linear in the size of the transcript of the original QSZK proof system. A similar improvement also applies to the upper bound for the non-interactive variant NIQSZK. Our main techniques are algorithmic versions of the Holevo-Helstrom measurement and the Uhlmann transform, both implementable in quantum linear space, implying polynomial-time complexity in the state dimension, using the recent space-efficient quantum singular value transformation of Le Gall, Liu, and Wang (CC, to appear).
Cite as
François Le Gall, Yupan Liu, and Qisheng Wang. A Slightly Improved Upper Bound for Quantum Statistical Zero-Knowledge. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 12:1-12:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{legall_et_al:LIPIcs.MFCS.2026.12,
author = {Le Gall, Fran\c{c}ois and Liu, Yupan and Wang, Qisheng},
title = {{A Slightly Improved Upper Bound for Quantum Statistical Zero-Knowledge}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {12:1--12:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.12},
URN = {urn:nbn:de:0030-drops-273933},
doi = {10.4230/LIPIcs.MFCS.2026.12},
annote = {Keywords: Quantum statistical zero-knowledge, Algorithmic Holevo-Helstrom measurement, Algorithmic Uhlmann transform}
}
Document
Algorithmic Information Bounds for Distances and Orthogonal Projections
Abstract
We introduce a new technique for proving bounds on the Kolmogorov complexity of geometric objects in Euclidean space, such as points and lines. We apply this technique to prove two theorems on algorithmic information theory, both of which have consequences for well-known problems in geometric measure theory. First, we show that for any point x in the plane and any other point y sufficiently independent of x, the distance between x and y retains at least half the complexity of the original point x. By the point-to-set principle of J. Lutz and N. Lutz, this yields an improved lower bound on the Hausdorff dimension of pinned distance sets, a topic closely related to Falconer’s distance set conjecture. Second, we prove an analogous result for orthogonal projections: for any point x in the plane and any line through the origin which is sufficiently independent of x, the projection of x onto that line retains at least half the complexity of x. As a consequence, we obtain a generalization of a theorem of Bourgain on exceptional sets for orthogonal projections.
Cite as
Peter Cholak, Marianna Csörnyei, Neil Lutz, Patrick Lutz, Elvira Mayordomo, and D. M. Stull. Algorithmic Information Bounds for Distances and Orthogonal Projections. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cholak_et_al:LIPIcs.MFCS.2026.13,
author = {Cholak, Peter and Cs\"{o}rnyei, Marianna and Lutz, Neil and Lutz, Patrick and Mayordomo, Elvira and Stull, D. M.},
title = {{Algorithmic Information Bounds for Distances and Orthogonal Projections}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {13:1--13:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.13},
URN = {urn:nbn:de:0030-drops-273941},
doi = {10.4230/LIPIcs.MFCS.2026.13},
annote = {Keywords: effective fractal dimensions, algorithmic randomness, Kolmogorov complexity}
}
Document
Beating Trivial Time for Tricky Triangle Tasks
Abstract
For several well-studied triangle detection problems in the literature, the trivial enumeration algorithms are known to be optimal (up to the exponent) assuming popular fine-grained conjectures. For example, All-Edges Sparse Triangle and Sparse Monochromatic Triangle where each node has degree n^δ for some δ < 1, and the Exact Triangle where edges have arbitrary weights, all have this property under the 3SUM Conjecture. However, as there are slightly nontrivial algorithms for 3SUM, it is natural to wonder if the trivial algorithm for these tricky triangle tasks might also be improved.
Applying a variety of techniques from randomized algorithms, circuit complexity, and communication complexity, we present the first improvements over the trivial algorithms for each of these problems in the Word RAM model. Moreover, our algorithms can be implemented with only polysize AC0 operations on words. Extending our techniques, we also show how to solve the notorious 4-cycle detection problem on n-node graphs in o(n²) time, in a Word-RAM model with word size w > ω(log² n). Along the way, we show how to sort n items over a universe of size 2^u using only AC0 word operations in O(n u log n)/w time.
Cite as
Neha Pant and Ryan Williams. Beating Trivial Time for Tricky Triangle Tasks. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{pant_et_al:LIPIcs.MFCS.2026.14,
author = {Pant, Neha and Williams, Ryan},
title = {{Beating Trivial Time for Tricky Triangle Tasks}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {14:1--14:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.14},
URN = {urn:nbn:de:0030-drops-273954},
doi = {10.4230/LIPIcs.MFCS.2026.14},
annote = {Keywords: sparse graph algorithms, triangle, Word RAM, 4-cycle}
}
Document
Bi-Criteria Approximations for Vertex Deletion Problems and d-Hitting Set
Abstract
We study bi-criteria approximation algorithms for vertex deletion problems in the (k,W) setting, where both the solution size and total weight are bounded simultaneously. Given a graph G, a weight function w:V → ℚ^+, a size bound k, and a weight budget W, a bi-criteria (a,b)-approximation algorithm either certifies that no solution of size at most k and weight at most W exists, or returns a solution of size at most ak and weight at most bW. Parameterizing by the solution size k - rather than the weight budget W - allows our algorithms to handle arbitrary positive rational weights without any lower bound assumption, addressing a fundamental limitation of prior W-parameterized approaches.
We obtain two families of results. For general vertex deletion problems Π-Deletion admitting a polynomial-time weighted α-approximation, we obtain a polynomial-time (α(λ+1),α(1+1/(λ)))-approximation for any λ > 0, a randomized FPT improvement for problems admitting a sampling step, and a deterministic FPT version for problems with bounded obstruction size. For (k,W)-d-Hitting Set, which captures vertex deletion problems with obstruction size at most d, we design a polynomial-time (d,d)-approximation, a parameterized family of ((1-ε)d, d)-approximations improving the size factor below d, and two algorithms that simultaneously push both factors below d: a ((d+1)/2,(d+1)/2)-approximation and a more refined (d-γ,d-γ)-approximation for any γ ∈ (0,(d-1)/2). All algorithms work with arbitrary positive rational weights and are parameterized by the solution size k.
To demonstrate the broad applicability of our framework, we instantiate our results on six well-studied vertex deletion problems: Cluster Vertex Deletion, FVS in Tournaments, Split Vertex Deletion, Feedback Vertex Set, d-Path Vertex Cover, and Pathwidth-One Vertex Deletion. In fact, our general results apply to any vertex deletion problem admitting a polynomial-time weighted approximation algorithm, and the six problems serve as representative examples spanning a range of obstruction structures - from bounded-size obstructions to unbounded ones. For (k,W) setting of Feedback Vertex Set and Pathwidth-One Vertex Deletion, we establish new sampling steps enabling the FPT approximation results. For Pathwidth-One Vertex Deletion, we additionally prove a polynomial-time 3-approximation for the weighted version on general graphs.
Cite as
Soumen Mandal, Ashutosh Rai, and Saket Saurabh. Bi-Criteria Approximations for Vertex Deletion Problems and d-Hitting Set. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{mandal_et_al:LIPIcs.MFCS.2026.15,
author = {Mandal, Soumen and Rai, Ashutosh and Saurabh, Saket},
title = {{Bi-Criteria Approximations for Vertex Deletion Problems and d-Hitting Set}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {15:1--15:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.15},
URN = {urn:nbn:de:0030-drops-273963},
doi = {10.4230/LIPIcs.MFCS.2026.15},
annote = {Keywords: Parameterized approximation algorithms, bi-criteria approximation, vertex deletion problems, d-Hitting Set, branching algorithms, sampling step}
}
Document
Bilateralism with Incompatible Proofs and Refutations
Abstract
Logical bilateralism challenges traditional concepts of logic by treating assertion and denial as independent yet opposed acts. While initially devised to justify classical logic, its constructive variants show that both acts admit intuitionistic interpretations. This paper presents a bilateral system where a formula cannot be both provable and refutable without contradiction, offering a framework for modelling mathematical proofs and refutations that exclude inconsistency. We formalise the logic via a bilateral natural deduction system with the desirable proof-theoretic properties of normalisation, subformula property and consistency, together with a base-extension semantics grounded in explicit proofs and refutations. Finally, refutation is shown to coincide with Nelson’s constructive falsity, extending intuitionistic logic for constructive epistemic reasoning.
Cite as
Victor Barroso-Nascimento, Maria Osório, and Elaine Pimentel. Bilateralism with Incompatible Proofs and Refutations. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{barrosonascimento_et_al:LIPIcs.MFCS.2026.16,
author = {Barroso-Nascimento, Victor and Os\'{o}rio, Maria and Pimentel, Elaine},
title = {{Bilateralism with Incompatible Proofs and Refutations}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {16:1--16:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.16},
URN = {urn:nbn:de:0030-drops-273974},
doi = {10.4230/LIPIcs.MFCS.2026.16},
annote = {Keywords: Proofs and refutations, Constructive falsity, Natural deduction, Logical bilateralism, Base-extension semantics}
}
Document
Boolean Combinations of ω-Rational Trace Languages: Emptiness, Rationality, Regularity
Abstract
This paper studies decision problems for Boolean combinations of ω-rational trace languages. The complexities of these decision problems (emptiness, regularity, rationality) are classified depending on properties of the independence alphabets and of the Boolean combinations allowed. For any of the problems, we obtain a trichotomy ranging from decidable over a low level of the arithmetical hierarchy to a low level of the analytical hierarchy.
Cite as
Dietrich Kuske. Boolean Combinations of ω-Rational Trace Languages: Emptiness, Rationality, Regularity. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 17:1-17:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kuske:LIPIcs.MFCS.2026.17,
author = {Kuske, Dietrich},
title = {{Boolean Combinations of \omega-Rational Trace Languages: Emptiness, Rationality, Regularity}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {17:1--17:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.17},
URN = {urn:nbn:de:0030-drops-273981},
doi = {10.4230/LIPIcs.MFCS.2026.17},
annote = {Keywords: Mazurkiewicz traces, rational and regular languages, degrees of unsolvability}
}
Document
Burning Graph Powers and Branching Trees
Abstract
Graph burning is a discrete-time process that models the spread of social contagion. Initially, all vertices are unburned. In each round, one unburned vertex is selected and burned, while any unburned vertex that has a burned neighbour from the previous round also becomes burned. The burning number of a graph is the minimum number of rounds needed to burn the entire graph. In this paper, we study the burning number of graph powers. First, we show that for a connected graph G, its graph power G^k contains a (k+1)^+-branching tree as a spanning tree. A (k+1)^+-branching tree is one in which all internal vertices have degree at least k+1. We then show that (k+1)^+-branching trees on n vertices have burning number at most ⌈√{4(k-1)n/k²}⌉. As the burning number of a graph is at most the burning number of any of its spanning trees, this gives an upper bound on the burning number of graph powers. We also derive an alternative upper bound on the burning number of k^+-branching trees using the strongest currently known general burning number bound [Bastide et al.]. We then identify the ranges of k and n for which our bound outperforms or matches this alternative bound. Finally, we show that b(G^k) ≤ (1+o(1))√{n/k} based on the asymptotic burning number bound of Norin and Turcotte.
Cite as
Jesper Jansson, Shashanka Kulamarva, Yukihiro Murakami, and Nikolaas Verhulst. Burning Graph Powers and Branching Trees. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 18:1-18:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{jansson_et_al:LIPIcs.MFCS.2026.18,
author = {Jansson, Jesper and Kulamarva, Shashanka and Murakami, Yukihiro and Verhulst, Nikolaas},
title = {{Burning Graph Powers and Branching Trees}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {18:1--18:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.18},
URN = {urn:nbn:de:0030-drops-273990},
doi = {10.4230/LIPIcs.MFCS.2026.18},
annote = {Keywords: Graph burning, Burning number, Graph power, k^+-branching tree}
}
Document
Characterizing LTL Formulas by Examples
Abstract
We investigate the extent to which Linear Temporal Logic (LTL) formulas can be uniquely characterized by a finite set of labeled examples. We consider different types of examples, ranging from finite words to transfinite words, as well as schematic examples. In the finite-word setting, we provide a complete classification of basis-restricted LTL fragments that admit such unique characterizations. Next, we show that allowing transfinite words as examples enables finite unique characterizations for large monotone fragments of LTL. Finally, we introduce schematic examples, i.e., patterns that compactly represent a family of finite words, and we show that these enable unique characterization results in the finite setting that were not possible with ordinary finite examples alone. Overall, the work provides a foundational account of the descriptive power of different example types for example-driven specification, debugging, and learning of temporal properties.
Cite as
Balder ten Cate, Dana Fisman, Roi Ohayon, and Patrik Sestic. Characterizing LTL Formulas by Examples. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 19:1-19:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{tencate_et_al:LIPIcs.MFCS.2026.19,
author = {ten Cate, Balder and Fisman, Dana and Ohayon, Roi and Sestic, Patrik},
title = {{Characterizing LTL Formulas by Examples}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {19:1--19:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.19},
URN = {urn:nbn:de:0030-drops-274000},
doi = {10.4230/LIPIcs.MFCS.2026.19},
annote = {Keywords: Linear Temporal Logic, Examples, Transfinite Words}
}
Document
Complexity of Clique-Guarded First-Order Logic with Counting
Abstract
We introduce clique-guarded first-order logic with counting (cgFOC), a fragment of the first-order logic with counting FOC [Kuske and Schweikardt, LICS 2017], and we study the complexity of this fragment. In particular, we prove computable upper bounds on the Vapnik-Chervonenkis (VC) dimension of cgFOC formulas and on the graph dimension of cgFOC counting terms on nowhere dense classes of relational structures. Furthermore, we show algorithmic metatheorems for cgFOC for query answering, enumeration, and probably approximately correct (PAC) learning for Boolean and multiclass classification problems on classes of locally bounded expansion. On the other hand, we show that a slight extension of cgFOC is already intractable on trees.
Cite as
Steffen van Bergerem, Johannes Friedrich Lange, and Nicole Schweikardt. Complexity of Clique-Guarded First-Order Logic with Counting. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{vanbergerem_et_al:LIPIcs.MFCS.2026.20,
author = {van Bergerem, Steffen and Lange, Johannes Friedrich and Schweikardt, Nicole},
title = {{Complexity of Clique-Guarded First-Order Logic with Counting}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {20:1--20:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.20},
URN = {urn:nbn:de:0030-drops-274012},
doi = {10.4230/LIPIcs.MFCS.2026.20},
annote = {Keywords: First-order logic with counting, VC dimension, graph dimension, algorithmic metatheorems, enumeration, nowhere dense, locally bounded expansion, PAC learning}
}
Document
Compression for Coinductive Rewriting and the Cut-Elimination of Non-Wellfounded Proofs
Abstract
We introduce a generic presentation of "syntactic objects built by mixed induction and coinduction" encompassing all standard kinds of infinitary terms, as well as derivation trees in non-wellfounded proof systems. We then define a coinductive notion of infinitary rewriting of such objects, which is equivalent to the original presentation of infinitary rewriting relying on metric convergence and ordinal-indexed sequences of rewriting steps. This provides a unified coinductive presentation of e.g. first-order infinitary rewriting, infinitary λ-calculi, and cut-elimination in non-wellfounded proofs.
We then formulate and study the coinductive counterpart of compression, i.e. the property of an infinitary rewriting system such that all rewriting sequences of any ordinal length can be "compressed" to equivalent sequences of length at most ω (which ensures that they can be finitely approximated). We characterise compression in our generic setting for coinductive rewriting, "factorising" the part of the proof that can be performed at this level of generality. Our proof is fully coinductive, avoiding any detour via rewriting sequences.
Finally we focus on the non-wellfounded proof system µMALL^∞ for multiplicative-additive linear logic with fixed points, and we put our results to work in order to prove that compression holds for cut-elimination in this setting, which is a key lemma of several extensions of cut-elimination to similar systems.
Cite as
Rémy Cerda and Alexis Saurin. Compression for Coinductive Rewriting and the Cut-Elimination of Non-Wellfounded Proofs. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 21:1-21:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cerda_et_al:LIPIcs.MFCS.2026.21,
author = {Cerda, R\'{e}my and Saurin, Alexis},
title = {{Compression for Coinductive Rewriting and the Cut-Elimination of Non-Wellfounded Proofs}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {21:1--21:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.21},
URN = {urn:nbn:de:0030-drops-274026},
doi = {10.4230/LIPIcs.MFCS.2026.21},
annote = {Keywords: coinduction, non-wellfounded proof theory, rewriting, compression lemma, lambda-calculus, cut-elimination}
}
Document
Computational Power of Energy-Constrained Autonomous Robots Under Sequential Schedulers
Abstract
We consider the distributed framework of swarms of mobile robots. A swarm is a set of computational, anonymous, indistinguishable, homogeneous, and autonomous entities that operate in the Euclidean plane through infinite sequences of Look-Compute-Move cycles. The goal of a swarm is to collaborate to solve a given problem. The ability to solve a problem depends on the swarm features and its setting X^S, where X ∈ {OBLOT, FSTA, FCOM, LUMI} denotes the memory/communication model and S denotes the class of schedulers (e.g., fully-synchronous, sequential, asynchronous) that activate the robots. Given a pool of settings, prior research has characterized the relations (dominance, equivalence, or orthogonality) among their computational powers, recently extending this analysis to the class of sequential schedulers (i.e., activating only one robot per round), and of the restricted ones (i.e., never activating a robot twice consecutively).
In this paper, we extend the study on sequential schedulers (SEQ, PERM, and RROBIN) by defining two classes of sequential restricted schedulers R-SEQ and R-PERM. In particular, we analyze how the computational power of each model OBLOT, LUMI, and FCOM is affected by considering both sequential schedulers and their restricted variants; for FSTA, we only provide the relation between RROBIN and R-PERM. We establish both equivalence and dominance results: some settings are computationally equivalent, while others can be separated by problems solvable in one setting but not in the other.
Cite as
Caterina Feletti, Paola Flocchini, and Nicola Santoro. Computational Power of Energy-Constrained Autonomous Robots Under Sequential Schedulers. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{feletti_et_al:LIPIcs.MFCS.2026.22,
author = {Feletti, Caterina and Flocchini, Paola and Santoro, Nicola},
title = {{Computational Power of Energy-Constrained Autonomous Robots Under Sequential Schedulers}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {22:1--22:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.22},
URN = {urn:nbn:de:0030-drops-274038},
doi = {10.4230/LIPIcs.MFCS.2026.22},
annote = {Keywords: Autonomous mobile robots, Look-Compute-Move, Computational power, Sequential schedulers, Energy-constrained}
}
Document
Connectivity Augmentation of Plane Graphs
Abstract
We study the problem of connectivity augmentation of a planar graph, while preserving planarity. This problem is motivated by many real-world settings such as road-networks, power-networks etc. In these settings, it is crucial to preserve the original planar embedding after augmentation.
In 2009, Gutwenger and Mutzel gave a constructive algorithm showing that a connected planar graph with a fixed embedding (a plane graph) can be optimally augmented to a biconnected graph without crossings while preserving the embedding. We further this line of research, by giving an algorithm that computes a minimum set of edges that makes a connected plane graph 2-edge-connected in O(|V|(1+α(|V|))) time and linear space, where α is the inverse Ackermann function.
We also study the 3-vertex-connectivity augmentation of biconnected outerplanar plane graphs. We present the first polynomial-time algorithm that augments such graphs to 3-connectivity with the minimum number of edges in O(|V|(1+α(|V|))) time and linear space while preserving the embedding, i.e. the augmented graph has a planar embedding that extends the given embedding.
Cite as
Krishnan Dehaleesan, Asif Khan, and Pranabendu Misra. Connectivity Augmentation of Plane Graphs. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 23:1-23:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dehaleesan_et_al:LIPIcs.MFCS.2026.23,
author = {Dehaleesan, Krishnan and Khan, Asif and Misra, Pranabendu},
title = {{Connectivity Augmentation of Plane Graphs}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {23:1--23:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.23},
URN = {urn:nbn:de:0030-drops-274044},
doi = {10.4230/LIPIcs.MFCS.2026.23},
annote = {Keywords: Connectivity augmentation, Plane graphs, Bridgetree, BC-tree, Balanced graph}
}
Document
Constant-Time Dynamic Enumeration of Word Infixes in a Regular Language
Abstract
For a fixed regular language L, the enumeration of L-infixes is the following task: we are given an input word w = a₁ ⋯ a_n and we must enumerate the infixes of w that belong to L, i.e., the pairs i ≤ j such that a_i ⋯ a_j ∈ L. We are interested in dynamic enumeration of L-infixes, where we must additionally support letter substitution updates on w (e.g., "replace the i-th letter of w by a letter a"). Each update changes the set of infixes to enumerate, and resets the enumeration state.
We study for which regular languages L we can perform dynamic enumeration of L-infixes in constant delay (i.e., the next infix is always produced in constant time) and constant additional memory throughout the enumeration, while supporting each update in constant time.
We show that, for languages L with a neutral letter, if the language L belongs to the class ZG and is extensible (i.e., if u ∈ L and u is a factor of v then v ∈ L), then dynamic enumeration of L-infixes can be achieved with a simple algorithm that ensures constant-time updates and constant delay, but not constant additional memory. Our main contribution is then to show an algorithm that additionally uses only constant additional memory, and applies to a more general class of semi-extensible ZG languages for which we give several equivalent characterizations. We further discuss whether our results can be generalized to larger language classes and show some (conditional) lower bounds.
Cite as
Antoine Amarilli, Sven Dziadek, and Luc Segoufin. Constant-Time Dynamic Enumeration of Word Infixes in a Regular Language. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 24:1-24:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{amarilli_et_al:LIPIcs.MFCS.2026.24,
author = {Amarilli, Antoine and Dziadek, Sven and Segoufin, Luc},
title = {{Constant-Time Dynamic Enumeration of Word Infixes in a Regular Language}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {24:1--24:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.24},
URN = {urn:nbn:de:0030-drops-274050},
doi = {10.4230/LIPIcs.MFCS.2026.24},
annote = {Keywords: regular language, dynamic membership, enumeration, infix, ZG, automata}
}
Document
Constructible Words Characterize Rational Languages of Words Indexed by Scattered Linear Orderings
Abstract
Automata on linear orderings are finite-state automata introduced by Bruyère and Carton as a broad generalization of finite, infinite and transfinite-word automata. In this context, a word is defined as a function from a linear ordering to a finite alphabet. This general definition can make automata on linear orderings difficult to reason about. In this work, we introduce constructible words as an intuitive way of tackling this difficulty. These words can be obtained by a finite number of applications of simple operators and thus admit a finite notation. We show that a rational language of words indexed by scattered (countable and uncountable) linear orderings is characterized by its constructible words. Our proof of this result relies on an interesting theorem of semigroup theory due to Colcombet. We expect this property to be useful in future theoretical developments about automata on scattered linear orderings.
Cite as
Thomas Braipson and Tom Clara. Constructible Words Characterize Rational Languages of Words Indexed by Scattered Linear Orderings. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 25:1-25:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{braipson_et_al:LIPIcs.MFCS.2026.25,
author = {Braipson, Thomas and Clara, Tom},
title = {{Constructible Words Characterize Rational Languages of Words Indexed by Scattered Linear Orderings}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {25:1--25:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.25},
URN = {urn:nbn:de:0030-drops-274062},
doi = {10.4230/LIPIcs.MFCS.2026.25},
annote = {Keywords: Automata on linear orderings, Rational languages, Ultimately periodic words, Constructible Words, Complementation, Algebraic properties of automata, Semigroups}
}
Document
Counting All Lattice Rectangles in the Square Grid in Near-Linear Time
Abstract
We study the exact counting problem for all lattice rectangles contained in the square [0,n)×[0,n), including non-axis-parallel ones. Starting from the standard parametrization by a primitive direction (u,v) and two side lengths, we derive a sequence of exact algorithms of complexity O(n²), O(n^{3/2} log n), O(n^{4/3} log n), and finally O(n log³n). The main idea behind the near-linear algorithm is to reduce the geometric summation to a constant-size family of weighted floor sums closed under Euclidean-style affine and reciprocal transformations, and hence evaluable in O(log n) time per query. The intermediate algorithms expose the structural reductions leading to this final kernel and provide independent cross-checks for the implementation.
Cite as
Dmitry Babichev and Sergey Babichev. Counting All Lattice Rectangles in the Square Grid in Near-Linear Time. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{babichev_et_al:LIPIcs.MFCS.2026.26,
author = {Babichev, Dmitry and Babichev, Sergey},
title = {{Counting All Lattice Rectangles in the Square Grid in Near-Linear Time}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {26:1--26:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.26},
URN = {urn:nbn:de:0030-drops-274076},
doi = {10.4230/LIPIcs.MFCS.2026.26},
annote = {Keywords: Lattice rectangles, grid enumeration, floor sums, M\"{o}bius inversion}
}
Document
Counting Equitable k-Colorings in Graphs of Bounded Clique-Width
Abstract
For a graph G, a proper k-coloring of G is equitable if the sizes of any two color classes differ by at most one. The Equitable k-Coloring problem asks, for a given graph G and integer k, whether G admits an equitable k-coloring. Bodlaender and Fomin (Theoretical Computer Science 2005) showed that it is polynomial-time solvable on graphs of bounded treewidth, while it remains NP-hard on cographs, and thus on graphs of constant clique-width. Fellows et al. (Information and Computation 2011) showed that the problem becomes W[1]-hard when parameterized by tree-width (and hence clique-width) plus the number of colors k.
We first show that, there exists an algorithm, given an integer k ≥ 1 and an n-vertex graph G together with a w-expression whose underlying unlabelled graph is G, computes the number of equitable k-colorings of G in time 2^O(k⋅w) ⋅ n^O(k). In particular, we show that for every fixed k, counting equitable k-colorings is polynomial-time solvable on graph classes of bounded clique-width, given a clique-width expression.
We then show that, under SETH, the dependence on clique-width in this algorithm is essentially optimal. As a consequence, our results provide a fairly tight picture of the complexity of Equitable k-Coloring with respect to the combined parameter k+clique-width in the following sense: For variable k, the problem is W[1]-hard, however for every fixed integer k, it is polynomial-time solvable on graphs of bounded clique-width given a clique-width expression, and this remains true even for the counting version.
Second, we refine our clique-width algorithm for the linear setting. We show that there exists an algorithm, given an integer k ≥ 1 and an n-vertex graph G together with a linear w-expression constructing G, computes the number of equitable k-colorings of G in time max{1,2^k-2}^w ⋅ n^{k+O(1)}. Thus, for bounded linear clique-width, we obtain a significantly sharper dependence on the width parameter than in the general clique-width case.
Third, we consider a different structural restriction, namely the class of P_t-free graphs. A graph is called P_t-free if it does not contain the path on t vertices as an induced subgraph. This is a different setting from bounded clique-width; in particular, already P₅-free graphs have unbounded clique-width. Nevertheless, we show that for every P_t-free graph G, the number of equitable list 3-colorings of G can be computed in subexponential time.
Cite as
Holger Dell, Thore Husfeldt, and Amir Nikabadi. Counting Equitable k-Colorings in Graphs of Bounded Clique-Width. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 27:1-27:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dell_et_al:LIPIcs.MFCS.2026.27,
author = {Dell, Holger and Husfeldt, Thore and Nikabadi, Amir},
title = {{Counting Equitable k-Colorings in Graphs of Bounded Clique-Width}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {27:1--27:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.27},
URN = {urn:nbn:de:0030-drops-274085},
doi = {10.4230/LIPIcs.MFCS.2026.27},
annote = {Keywords: Equitable coloring, Clique-width, P\underlinet-free graphs}
}
Document
Counting Patterns in Degenerate Graphs in Constant Space
Abstract
For a fixed pattern graph, we study the algorithmic complexity of counting homomorphisms, subgraph isomorphisms, and induced subgraph isomorphisms into an n-vertex, d-degenerate host graph. Bressan (Algorithmica, 2021) introduced the notion of DAG treewidth and showed that counting homomorphisms and induced subgraphs can be performed efficiently using dynamic programming that requires polynomial space. In this work, we introduce a new graph parameter, called DAG treedepth, which enables efficient divide-and-conquer algorithms for counting homomorphisms in d-degenerate host graphs using only constant space.
Bera, Gishboliner, Levanzov, Seshadhri, and Shapira (SODA, 2021) showed that a pattern graph has DAG treewidth one if and only if it contains no induced cycle of length at least six. This induced minor characterization leads to linear-time and linear-space algorithms. Building on this line of work, we derive an induced-minor characterization of graphs with DAG treedepth at most two that uses only constant space.
Recently, Paul-Pena and Seshadhri (ICALP, 2025) proved that all pattern graphs on at most nine vertices can be counted in subquadratic time using polynomial space. We show that every pattern graph on at most nine vertices can be counted as an induced subgraph in O(n³) time using only constant space. Moreover, we show that patterns on at most eleven vertices can be counted in O(n²) time using polynomial space.
Finally, we present a constant-space algorithm for counting induced subgraphs that matches the running time of Bressan’s algorithm. We further show that, when polynomial space is allowed, homomorphisms, subgraph isomorphisms, and induced subgraph isomorphisms can be counted faster than Bressan’s algorithm. In addition, we establish several other results related to DAG treewidth and DAG treedepth that may be of independent interest.
Cite as
Balagopal Komarath, Anant Kumar, and Akash Pareek. Counting Patterns in Degenerate Graphs in Constant Space. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 28:1-28:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{komarath_et_al:LIPIcs.MFCS.2026.28,
author = {Komarath, Balagopal and Kumar, Anant and Pareek, Akash},
title = {{Counting Patterns in Degenerate Graphs in Constant Space}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {28:1--28:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.28},
URN = {urn:nbn:de:0030-drops-274091},
doi = {10.4230/LIPIcs.MFCS.2026.28},
annote = {Keywords: Homomorphism Counting, Subgraph Counting, Induced subgraph Counting, Bounded degeneracy graph}
}
Document
Deciding the Common Fragment of CTL with past and LTL
Abstract
A central goal of language theory is to compare formalisms by understanding both their expressive overlaps and their relative expressive power. One particularly challenging question in this direction is the problem of determining the common fragment of two formalisms F₁ and F₂, that is, effectively characterise the class F₁∩ F₂ of properties that can be expressed in both formalisms. This question can be equally phrased as a decision problem: given a property expressed in F₁ or F₂, decide whether the same property can be also expressed in F₁∩ F₂. A question closely related to this is the membership problem, denoted F₁ ↦ F₂, which asks whether a property expressed in F₁ can be also expressed in F₂. These problems become particularly difficult when branching-time formalisms are involved, in general due to the lack of equivalent algebraic characterizations.
In this work, we prove that LTL ∩ PCTL is decidable, where PCTL denotes CTL extended with past operators. We do this by showing that both membership problems, LTL ↦ PCTL and PCTL ↦ LTL, are decidable. The direction PCTL ↦ LTL follows from suitable combinations of known results. The converse direction, LTL ↦ PCTL, requires an automata-theoretic characterisation of PCTL. Specifically, we introduce a new class of automata, called counter-free hesitant weak tree automata (HWT_cf) that capture precisely the expressiveness of PCTL, and that are obtained by combining two orthogonal restrictions on alternating parity tree automata, namely, counter-free hesitancy and weakness. We then prove that, for every word language L defined by an LTL formula, the associated tree language △[L] is recognisable by an HWT_cf if and only if L is recognized by a deterministic Büchi word automaton. Since the latter recognisability problem is known to be decidable, so is the former.
This result advances the longstanding open problem of deciding LTL ∩ CTL. Indeed, that problem can now be reduced to PCTL ↦ CTL, that is, the question of when past operators can be eliminated.
Cite as
Massimo Benerecetti, Dario Della Monica, Angelo Matteo, Fabio Mogavero, and Gabriele Puppis. Deciding the Common Fragment of CTL with past and LTL. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 29:1-29:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{benerecetti_et_al:LIPIcs.MFCS.2026.29,
author = {Benerecetti, Massimo and Della Monica, Dario and Matteo, Angelo and Mogavero, Fabio and Puppis, Gabriele},
title = {{Deciding the Common Fragment of CTL with past and LTL}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {29:1--29:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.29},
URN = {urn:nbn:de:0030-drops-274104},
doi = {10.4230/LIPIcs.MFCS.2026.29},
annote = {Keywords: Membership problems, tree languages, tree logics, tree automata, Monadic Path Logic, CTL^*, CTL with past}
}
Document
Distinguishing Elements in Semigroups
Abstract
We investigate randomized streaming algorithms for word problems in finitely generated semigroups. For this we use the notion of a distinguisher: a randomized streaming algorithm that processes two input words in parallel and, with high probability, reaches identical memory states if the words represent the same element, and distinct states otherwise. We construct such distinguishers with space complexity 𝒪(log log n) for finitely generated commutative semigroups. Moreover, we show a transfer result for semilattice decompositions that allows to construct a distinguisher for a finitely generated semigroup from distinguishers for the components of its semilattice decomposition. Thereby the space complexity and the error probability of the distinguisher increase only by a constant factor. We use this result to obtain distinguishers with space complexity 𝒪(log n) for free Clifford semigroups and distinguishers with space complexity 𝒪(log log n) for finitely generated regular nilpotent semigroups. We complement these upper bounds with lower bounds demonstrating that certain well-known semigroups do not admit distinguishers with sublinear space complexity. This includes, for example, free inverse monoids of rank at least two and polycyclic semigroups.
Cite as
Markus Lohrey, Alexander Thumm, and Julio Xochitemol. Distinguishing Elements in Semigroups. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 30:1-30:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{lohrey_et_al:LIPIcs.MFCS.2026.30,
author = {Lohrey, Markus and Thumm, Alexander and Xochitemol, Julio},
title = {{Distinguishing Elements in Semigroups}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {30:1--30:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.30},
URN = {urn:nbn:de:0030-drops-274114},
doi = {10.4230/LIPIcs.MFCS.2026.30},
annote = {Keywords: Streaming algorithms, semigroups, word problem, space complexity}
}
Document
Efficient Algorithms for the Bottleneck Path Problem in Geometric Graphs
Abstract
We present efficient algorithms for the bottleneck path problem in two geometric settings that arise naturally in applications: directional-antenna graphs in the plane with antenna angles bounded from below by a constant, and visibility graphs whose vertices lie on or above a 1.5-dimensional terrain, both with Euclidean distances as edge weights. We provide near-linear algorithms for the corresponding decision problems, namely, determining whether the subgraph obtained by retaining all edges with weight at most some threshold bn contains a path from s to t. We then use the decision procedures to obtain algorithms for the bottleneck path problem that run in O^*(n^{8/7}) randomized expected time, where n is the input size and the O^*(⋅) notation hides subpolynomial factors.
Within the same performance bounds, we can also solve the bounded-hop version, in which we only consider s-t paths with at most k edges, for a given integer k < n.
Cite as
Matthew J. Katz, Rachel Saban, and Micha Sharir. Efficient Algorithms for the Bottleneck Path Problem in Geometric Graphs. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 31:1-31:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{katz_et_al:LIPIcs.MFCS.2026.31,
author = {Katz, Matthew J. and Saban, Rachel and Sharir, Micha},
title = {{Efficient Algorithms for the Bottleneck Path Problem in Geometric Graphs}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {31:1--31:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.31},
URN = {urn:nbn:de:0030-drops-274121},
doi = {10.4230/LIPIcs.MFCS.2026.31},
annote = {Keywords: Bottleneck path, BFS, Antennas, Terrain, Visibility, Bichromatic closest pair, Bounded hop, Reverse shortest paths}
}
Document
En Route to a Standard QMA₁ vs. QCMA Oracle Separation
Abstract
We study the power of quantum witnesses under perfect completeness. We construct a classical oracle relative to which a language lies in QMA₁ but not in QCMA when the QCMA verifier is only allowed polynomially many adaptive rounds and exponentially many parallel queries per round. Additionally, we derandomize the permutation-oracle separation of Fefferman and Kimmel, obtaining an in-place oracle separation between QMA₁ and QCMA. Furthermore, we focus on QCMA and QMA with an exponentially small gap, where we show a separation assuming the gap is fixed, but not when it may be arbitrarily small. Finally, we derive consequences for approximate ground-state preparation from sparse Hamiltonian oracle access, including a bounded adaptivity frustration-free variant.
Cite as
David Miloschewsky, Supartha Podder, and Dorian Rudolph. En Route to a Standard QMA₁ vs. QCMA Oracle Separation. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 32:1-32:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{miloschewsky_et_al:LIPIcs.MFCS.2026.32,
author = {Miloschewsky, David and Podder, Supartha and Rudolph, Dorian},
title = {{En Route to a Standard QMA₁ vs. QCMA Oracle Separation}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {32:1--32:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.32},
URN = {urn:nbn:de:0030-drops-274139},
doi = {10.4230/LIPIcs.MFCS.2026.32},
annote = {Keywords: Quantum complexity theory, Quantum Merlin-Arthur (QMA)}
}
Document
Eve-Positional Languages: Putting Order into Büchi Automata
Abstract
An ω-regular language is Eve-positional if, in all games with this language as objective, the existential player can play optimally without keeping any information from the previous moves. This notion plays a crucial role in verification, automata theory and synthesis.
Casares and Ohlmann recently gave several characterisations of Eve-positionality of ω-regular languages. For this, they introduce the notion of ε-complete parity automaton and show (among other results) that an ω-regular language is Eve-positional if and only if it can be recognised by some ε-completion of a deterministic parity automaton. Colcombet and Idir built on their work, and obtained a more direct algebraic characterisation of Eve-positionality.
We introduce a new formalism that characterises the Eve-positional languages, consisting of a restriction of non-deterministic Büchi automata. This allows us to complete a missing implication in Casares and Ohlmann’s work. We then use this formalism to describe a determinization procedure for non-deterministic Büchi automata recognising such languages, with size blow-up at most factorial. We also show that this construction is state-wise optimal for languages over sufficiently complete alphabets.
Cite as
Olivier Idir. Eve-Positional Languages: Putting Order into Büchi Automata. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 33:1-33:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{idir:LIPIcs.MFCS.2026.33,
author = {Idir, Olivier},
title = {{Eve-Positional Languages: Putting Order into B\"{u}chi Automata}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {33:1--33:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.33},
URN = {urn:nbn:de:0030-drops-274142},
doi = {10.4230/LIPIcs.MFCS.2026.33},
annote = {Keywords: B\"{u}chi automata, parity automata, Eve-positional language, \epsilon-complete automata, positional strategy, ordered B\"{u}chi automata}
}
Document
Exact Cut Complexity of Equal-Length Proportional Cake Cutting
Abstract
We study proportional cake cutting on the interval [0,1] under an equal-length constraint requiring each of the n agents to receive a bundle of length exactly 1/n and to assign value at least 1/n to that bundle. We determine the exact worst-case cut complexity of this problem. The exact value is 2n-2 cuts for every n ≥ 1. The lower bound follows from a simple identical-valuation instance, and the main contribution is the matching upper bound, since the only all-n upper bound previously available for this problem was quadratic. Our upper-bound proof starts from the constrained necklace-splitting theorem of Jojić, Panina, and Živaljević, which gives the required partition into equal-length bundles when the number of bundles is a prime power. The main difficulty is to convert this prime-power input into an exact all-n cut bound while preserving the equal-length constraint. When r is a prime-power divisor of n and s = n/r, our transfer principle constructs r equal-length bundles, builds a balanced fractional assignment of agents to bundles, rounds it by Hall’s theorem to an assignment in which each bundle receives exactly s agents, and recurses inside the bundles without additional overhead beyond the recursive cuts. Using the same constrained necklace-splitting theorem, we also show that 2n-2 cuts suffice for equal-length envy-freeness when n is a prime power. For all n, we give an O(n^1.525) upper bound via a peeling argument based on the Stromquist-Woodall exact-share theorem. The exact all-n envy-free cut complexity remains open.
Cite as
Yasushi Kawase and Mohammad Azharuddin Sanpui. Exact Cut Complexity of Equal-Length Proportional Cake Cutting. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 34:1-34:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kawase_et_al:LIPIcs.MFCS.2026.34,
author = {Kawase, Yasushi and Sanpui, Mohammad Azharuddin},
title = {{Exact Cut Complexity of Equal-Length Proportional Cake Cutting}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {34:1--34:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.34},
URN = {urn:nbn:de:0030-drops-274158},
doi = {10.4230/LIPIcs.MFCS.2026.34},
annote = {Keywords: cake cutting, fair division, cut complexity, proportionality, envy-freeness}
}
Document
Fast Rational Search via Stern-Brocot Tree
Abstract
We revisit the problem of rational search: given an unknown rational number α = a/b ∈ (0,∞) with a,b ≤ n, the goal is to identify α using comparison queries of the form "β ≤ α?". The problem has been studied several decades ago and optimal query algorithms are known. We present an algorithm for rational search based on a compressed traversal of the Stern-Brocot tree, which appeared to have been overlooked in the literature. This approach also naturally extends to two related problems that, to the best of our knowledge, have not been previously addressed: (i) unbounded rational search, where the bound n is unknown, and (ii) computing the best (in a precise sense) rational approximation of an unknown real number using only comparison queries. While the algorithm is simple and natural, one of our main contributions is its analysis: we give an upper and lower bound on its worst-case query complexity.
Cite as
Connor Weyers and N. V. Vinodchandran. Fast Rational Search via Stern-Brocot Tree. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 35:1-35:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{weyers_et_al:LIPIcs.MFCS.2026.35,
author = {Weyers, Connor and Vinodchandran, N. V.},
title = {{Fast Rational Search via Stern-Brocot Tree}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {35:1--35:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.35},
URN = {urn:nbn:de:0030-drops-274169},
doi = {10.4230/LIPIcs.MFCS.2026.35},
annote = {Keywords: Rational number search, Continued fractions, Stern-Brocot tree, Rational approximation}
}
Document
Finding b-Colorings Using Feedback Edges
Abstract
A b-coloring of a graph is a proper vertex coloring such that each color class contains a vertex that sees all other colors in its neighborhood. The b-coloring problem, in which the task is to decide whether a graph admits a b-coloring with k colors, is NP-complete in general but polytime solvable on trees. Moreover, it is known that b-coloring is in XP but W[t]-hard for all t ∈ ℕ when parameterized by tree-width. In fact, only very few parameters, such as the vertex cover number, were known to admit an FPT algorithm for b-coloring.
In this paper, we consider a more restrictive parameter measuring similarity to trees than tree-width, namely the feedback edge number, and show that b-coloring is fixed-parameter tractable under this parameterization. Our algorithm combines standard techniques used in parameterized algorithmics with the problem-specific ideas used in the polytime algorithm for trees. In addition, we present an FPT algorithm for b-coloring parameterized by distance to co-cluster, which is a parameter measuring similarity to complete multipartite graphs. Finally, we make several observations based on known results, including that b-coloring is W[1]-hard when parameterized by tree-depth.
Cite as
Jakub Balabán. Finding b-Colorings Using Feedback Edges. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 36:1-36:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{balaban:LIPIcs.MFCS.2026.36,
author = {Balab\'{a}n, Jakub},
title = {{Finding b-Colorings Using Feedback Edges}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {36:1--36:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.36},
URN = {urn:nbn:de:0030-drops-274172},
doi = {10.4230/LIPIcs.MFCS.2026.36},
annote = {Keywords: b-coloring, fixed-parameter algorithms, feedback edge number, distance to co-cluster}
}
Document
Finding Shortest Reconfiguration Sequences on Independent Set Polytopes
Abstract
We initiate the study of the shortest reconfiguration problem for independent sets under the adjacency relation derived from the independent set polytope. Given a graph and two independent sets, the problem asks for a shortest sequence transforming one into the other such that the subgraph induced by the symmetric difference of any two consecutive sets is connected. This is equivalent to finding a shortest path on the 1-skeleton of the independent set polytope. We prove that the problem is NP-hard even on planar graphs of bounded degree, as well as on split graphs. Notably, the hardness for planar graphs of bounded degree still holds even when deciding whether the target can be reached in at most two steps. For split graphs, we further show the W[2]-hardness when parameterized by the number of steps, as well as the inapproximability of the optimal length. As a consequence, we prove that the length of a shortest path between two vertices of a 0/1 polytope in ℝⁿ described by O(n) linear inequalities is hard to approximate within a factor of (1-ε)ln n for any constant ε > 0, unless P = NP. On the positive side, we provide polynomial-time algorithms for block graphs, cographs, and bipartite chain graphs. Moreover, for paths and cycles, we show that the optimal length of the shortest reconfiguration sequence exactly matches a trivial upper bound.
Cite as
Jean Cardinal, Kevin Mann, Akira Suzuki, Takahiro Suzuki, Yuma Tamura, and Xiao Zhou. Finding Shortest Reconfiguration Sequences on Independent Set Polytopes. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 37:1-37:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cardinal_et_al:LIPIcs.MFCS.2026.37,
author = {Cardinal, Jean and Mann, Kevin and Suzuki, Akira and Suzuki, Takahiro and Tamura, Yuma and Zhou, Xiao},
title = {{Finding Shortest Reconfiguration Sequences on Independent Set Polytopes}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {37:1--37:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.37},
URN = {urn:nbn:de:0030-drops-274180},
doi = {10.4230/LIPIcs.MFCS.2026.37},
annote = {Keywords: combinatorial reconfiguration, independent set, combinatorial shortest path, NP-completeness}
}
Document
Finite-State Dimension and the Davenport-Erdős Theorem
Abstract
A 1952 result of Davenport and Erdős states that if p is an integer-valued polynomial, then the real number 0.p(1)p(2)p(3)… is Borel normal in base ten. A later result of Nakai and Shiokawa extends this result to polynomials with arbitrary real coefficients and all bases b ≥ 2. It is well-known that finite-state dimension, a finite-state effectivization of the classical Hausdorff dimension, characterizes the Borel normal sequences as precisely those sequences of finite-state dimension 1. For an infinite set A of natural numbers, and a base b ≥ 2, the base-b Copeland-Erdős sequence of A, CE_b(A), is the infinite sequence obtained by concatenating the base-b expansions of the numbers in A in increasing order. In this work we investigate the possible relationships between the finite-state dimensions of CE_b(A) and CE_b(p(A)) where p is a polynomial. We show that, if the polynomial is permitted to have arbitrary real coefficients, then for any s,s^′ in the unit interval, there is a set A of natural numbers and a linear polynomial p so that the finite-state dimensions of CE_b(A) and CE_b(p(A)) are s and s^′ respectively. The corresponding result for strong finite-state dimension is also shown. We demonstrate that linear polynomials with rational coefficients do not change the finite-state dimension of any Copeland-Erdős sequence, but there exist polynomials with rational coefficients of every larger integer degree that change the finite-state dimension of some sequence. We also prove the surprising fact that there exist sets A and integer-valued monomials p such that CE_b(A) is normal, but CE_b(p(A)) has finite-state dimension strictly less than one.
Cite as
Joe Clanin and Matthew Rayman. Finite-State Dimension and the Davenport-Erdős Theorem. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 38:1-38:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{clanin_et_al:LIPIcs.MFCS.2026.38,
author = {Clanin, Joe and Rayman, Matthew},
title = {{Finite-State Dimension and the Davenport-Erd\H{o}s Theorem}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {38:1--38:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.38},
URN = {urn:nbn:de:0030-drops-274195},
doi = {10.4230/LIPIcs.MFCS.2026.38},
annote = {Keywords: Normal numbers, finite-state dimension, polynomials}
}
Document
Finite-State Dimension for Continued Fractions: Betting, Entropy and Normality
Abstract
Finite-state dimension quantifies the asymptotic density of information in an infinite sequence as seen by finite automata, and can be viewed as a bounded-memory analogue of Hausdorff dimension. The theory is by now well developed for base-b expansions. In this paper we initiate the study of finite-state dimension in the setting of continued fractions. This setting brings several new difficulties. The natural reference measure is the Gauss measure, which is the canonical invariant probability measure for the Gauss transformation governing the continued fraction shift. Unlike in the base-b setting, however, this measure is not a product measure. In addition, the digit space is inherently infinite, so the symbolic setting is markedly less rigid than the usual finite-alphabet framework. Continued fraction analogues of effective dimension have received considerable attention in recent years, but a finite-state theory in this setting has so far been missing.
Finite-state dimension in the classical setting admits several equivalent formulations. We begin from the original finite-state s-gale viewpoint. In the continued fraction setting, however, the betting odds are no longer fixed in advance: unlike in the base-b case, the fair payoff for the next symbol is determined by conditional Gauss probabilities, and these vary with the current continued fraction cylinder. This makes the choice of a finite-state betting model genuinely nontrivial. We first examine a natural local finite-state betting model for truncated continued fraction digits, in which a gambler may use both the visible local context and an additional finite internal memory. We show that this model is too strong: there is a fixed finite-state gambler that wins at a positive exponential rate on every continued fraction normal point. Consequently, full dimension in this model does not characterize continued fraction normality, and the classical Schnorr-Stimm dichotomy fails.
We then isolate the source of this failure and introduce a restricted context-gambler model in which the gambler is allowed to use only the visible truncated context, with no additional hidden finite-state memory. For this restricted model, we obtain, in direct analogy with the base-b setting, an entropy-rate characterization of finite-state dimension in terms of conditional entropy rates, and from this derive an exact finite-state characterization of continued fraction normality: an irrational real has finite-state dimension 1 if and only if its continued fraction expansion is normal. Finally, we prove that the Schnorr-Stimm dichotomy does hold in this restricted setting: on a continued fraction normal point every such gambler either preserves constant capital or loses at an exponential rate, while on every non-normal point some such gambler wins at an exponential rate.
Cite as
Satyadev Nandakumar, Subin Pulari, and Akhil S. Finite-State Dimension for Continued Fractions: Betting, Entropy and Normality. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 39:1-39:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{nandakumar_et_al:LIPIcs.MFCS.2026.39,
author = {Nandakumar, Satyadev and Pulari, Subin and S, Akhil},
title = {{Finite-State Dimension for Continued Fractions: Betting, Entropy and Normality}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {39:1--39:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.39},
URN = {urn:nbn:de:0030-drops-274208},
doi = {10.4230/LIPIcs.MFCS.2026.39},
annote = {Keywords: Finite-state dimension, continued fractions, normality, finite-state gamblers, entropy rates}
}
Document
Flood-It with Jewelry - Characterizing the Game Complexity for Cograph Generalizations
Abstract
Flood-It is a single-player game played on a precolored graph G, where the objective is to make G monochromatic using as few flooding moves as possible. In each move, a color c is selected and all vertices reachable from a fixed pivot vertex via a monochromatic path are recolored with c. In the free variant, the pivot may be chosen anew in every move.
Deciding whether a graph can be made monochromatic in at most k moves is NP-complete for both variants, fixed and free. This hardness persists even under strong structural restrictions such as split graphs and trees. The Free Flood-It variant is generally considered more difficult than its fixed-pivot counterpart, as it remains hard on several graph classes where the latter becomes tractable, including co-comparability and AT-free graphs.
Cographs, that is, P₄-free graphs, are among the few classes on which even Free Flood-It is solvable in polynomial time and therefore serve as our starting point. We consider the ten natural one-vertex extensions of P₄ - referred to as jewels - and study the complexity of both flooding games on the 1024 graph classes obtained by forbidding subsets of these graphs as induced subgraphs.
Our main contribution is a polynomial-time algorithm for Free Flood-It on graphs that are free of the three jewels bull, gem, and P₅, covering 128 of the 1024 classes. In addition, we prove that both variants remain NP-complete on thin-spider graphs, which exclude the eight jewels banner, co-banner, chair, gem, house, kite, P₅, and C₅, thereby establishing hardness for 256 additional classes.
Combined with known algorithms and hardness results, our work determines the complexity of both Flood-It variants for 896 of the 1024 considered graph classes.
Cite as
Martin Darmüntzel, Christian Rosenke, and Mark Scheibner. Flood-It with Jewelry - Characterizing the Game Complexity for Cograph Generalizations. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 40:1-40:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{darmuntzel_et_al:LIPIcs.MFCS.2026.40,
author = {Darm\"{u}ntzel, Martin and Rosenke, Christian and Scheibner, Mark},
title = {{Flood-It with Jewelry - Characterizing the Game Complexity for Cograph Generalizations}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {40:1--40:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.40},
URN = {urn:nbn:de:0030-drops-274216},
doi = {10.4230/LIPIcs.MFCS.2026.40},
annote = {Keywords: Flood-It, Free Flood-It, cograph generalizations, polynomial time algorithms, NP-completeness}
}
Document
Forbidden Subgraph Problems with Predictions
Abstract
In the Online Delayed Connected H-Node-Deletion Problem, an unweighted graph is revealed vertex by vertex and it must remain free of any induced copies of a specific connected induced forbidden subgraph H at each point in time. To achieve this, an algorithm must, upon each occurrence of H, identify and irrevocably delete one or more vertices. The objective is to delete as few vertices as possible. We provide tight bounds on the competitive ratio for forbidden subgraphs H that do not contain true twins or that do not contain false twins.
We further consider the problem within the model of predictions, where the algorithm is provided with a single bit of advice for each revealed vertex. These predictions are considered to be provided by an untrusted source and may be incorrect. We present a family of algorithms solving the problem with predictions and show that it is Pareto-optimal with respect to consistency and robustness for the online vertex cover problem, for 2-connected forbidden subgraphs that do not contain true twins or that do not contain false twins, as well as for any forbidden path of length at least five.
Cite as
Hans-Joachim Böckenhauer, Melvin Jahn, Dennis Komm, and Moritz Stocker. Forbidden Subgraph Problems with Predictions. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 41:1-41:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bockenhauer_et_al:LIPIcs.MFCS.2026.41,
author = {B\"{o}ckenhauer, Hans-Joachim and Jahn, Melvin and Komm, Dennis and Stocker, Moritz},
title = {{Forbidden Subgraph Problems with Predictions}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {41:1--41:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.41},
URN = {urn:nbn:de:0030-drops-274226},
doi = {10.4230/LIPIcs.MFCS.2026.41},
annote = {Keywords: online node deletion, competitive ratio, forbidden subgraphs, predictions}
}
Document
Forwarding Packets Greedily on the Line
Abstract
We consider the problem of forwarding packets arriving online with their destinations in a line network. In each time step, each router can forward one packet along the edge to its right, and the packet arrives at the next router one time step later. Packets are forwarded until they reach their destination. The flow time of a packet is the elapsed time between its release and its arrival at its destination. The goal is to minimize the maximum flow time.
This problem was introduced by Antoniadis et al. in 2014, with a focus on line networks. They proposed several natural algorithms. For one, they proved that it is not O(1)-competitive; for others, they claimed analogous lower bounds, seemingly leaving no natural candidate for an O(1)-competitive algorithm.
In this paper, we study a natural algorithm not considered in that work. Our algorithm, simply called Greedy, selects packets according to their projected flow time under the assumption that they are not delayed any further. We focus on the special case in which each packet needs to be forwarded by one or two routers; this case captures core difficulties. We show that Greedy achieves a competitive ratio of exactly 2-2^{1-k}, where k is the number of active routers in the network.
We also give the first nontrivial general lower bound, which applies even to randomized algorithms: using the same type of instances as in our lower bound for Greedy, we show that no algorithm can be (4/3-ε)-competitive for any ε > 0.
Cite as
Joan Boyar, Lene M. Favrholdt, Kim S. Larsen, Kevin Schewior, and Rob van Stee. Forwarding Packets Greedily on the Line. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 42:1-42:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{boyar_et_al:LIPIcs.MFCS.2026.42,
author = {Boyar, Joan and Favrholdt, Lene M. and Larsen, Kim S. and Schewior, Kevin and van Stee, Rob},
title = {{Forwarding Packets Greedily on the Line}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {42:1--42:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.42},
URN = {urn:nbn:de:0030-drops-274230},
doi = {10.4230/LIPIcs.MFCS.2026.42},
annote = {Keywords: Online algorithms, Packet scheduling, Greedy algorithm}
}
Document
Freeze-Tag with Return
Abstract
In the standard Freeze-Tag Problem (FTP), an initially awake robot (the source) is in charge of waking up a swarm of sleeping robots by moving towards them, given that all the awake robots can participate in the awakening process. The goal is to minimize the makespan to wake up all robots assuming they move at unit speed. In this paper we introduce the Freeze-Tag-with-Return Problem (FTRP) variant, where the robots must eventually return to their initial positions.
In the Euclidean plane with n sleeping robots lying on the unit disk centered at the initial position of the source, we show a non-trivial relationship between FTP and FTRP by proving that the difference between the optimal makespan of both problems never exceeds 1.959, and is at least 1.732 in the worst-case. We also present several upper and lower bounds on the optimal makespan. In particular, we show that if the sleeping robots are in convex positions, then the optimal makespan is at most 2 + 2√2, which is achieved by some instances.
From an algorithmic point-of-view, we present single-exponential algorithms for general distance functions. In metric spaces, these algorithms are asymptotically optimal under the ETH, which we show via an NP-hardness reduction on unweighted graphs.
Cite as
Nicolas Bonichon, Cyril Gavoille, Nicolas Hanusse, Gabriel Le Bouder, Taïssir Marcé, and Nils Morawietz. Freeze-Tag with Return. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 43:1-43:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bonichon_et_al:LIPIcs.MFCS.2026.43,
author = {Bonichon, Nicolas and Gavoille, Cyril and Hanusse, Nicolas and Le Bouder, Gabriel and Marc\'{e}, Ta\"{i}ssir and Morawietz, Nils},
title = {{Freeze-Tag with Return}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {43:1--43:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.43},
URN = {urn:nbn:de:0030-drops-274241},
doi = {10.4230/LIPIcs.MFCS.2026.43},
annote = {Keywords: Freeze-Tag Problem, sleeping robots, metric spaces}
}
Document
Functorial Semantics for First-Order Theories
Abstract
Building on the recent axiomatisation of first-order bicategories, we develop a functorial semantics approach to the model theory of first-order logic. First-order theories 𝕋 are captured by free first-order bicategories ℱ_𝕋 and models of𝕋 are structure-preserving functors from ℱ_𝕋 to a first-order bicategory 𝐂. Elementary morphisms of models arise as lax natural transformations between such functors, and the classical Tarski-Vaught test and downward Löwenheim-Skolem theorem admit direct diagrammatic proofs. Our results instantiate classically when 𝐂 = Rel and hold uniformly for models valued in Rel(𝐃) over an arbitrary Boolean geometric category 𝐃 in which regular epis split.
Cite as
Filippo Bonchi, Alessandro Di Giorgio, Roberto Di Virgilio, and Paweł Sobociński. Functorial Semantics for First-Order Theories. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 44:1-44:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bonchi_et_al:LIPIcs.MFCS.2026.44,
author = {Bonchi, Filippo and Di Giorgio, Alessandro and Di Virgilio, Roberto and Soboci\'{n}ski, Pawe{\l}},
title = {{Functorial Semantics for First-Order Theories}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {44:1--44:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.44},
URN = {urn:nbn:de:0030-drops-274255},
doi = {10.4230/LIPIcs.MFCS.2026.44},
annote = {Keywords: First-order logic, Model theory, Functorial semantics, String diagrams}
}
Document
Generalized Snarks, Disjoint Perfect Matchings, and Graph Covers
Abstract
We explore the interplay among three classical notions in graph theory: edge-colorings, perfect matchings, and graph coverings (locally bijective homomorphisms of graphs). In this paper, we consider undirected graphs in full generality of this notion: in contrast to the standard notion of a simple graph, our graphs may contain loops, semi-edges, and multiple edges. Many well-studied graph concepts, including matchings, edge-colorings, and covering projections, extend naturally to such graphs. Nevertheless, the role of simple graphs for graph covering problems is central, as emphasized in [J. Bok, J. Fiala, N. Jedličková, J. Kratochvíl, and M. Seifrtová. Computational complexity of covering disconnected multigraphs. Discret. Appl. Math., 359:229–243, 2024]. In that work, a relation "being stronger" was defined (a graph A is stronger than a graph B if every simple graph that covers A also covers B), and it was conjectured that if A has no semi-edges, then A is stronger than B if and only if A covers B. In their extended abstract presented at Eurocomb'23, Kratochvíl and Nedela proved this conjecture for 3-regular 1-vertex graphs B (and arbitrary A). They also introduced the notion (A,B)-snark for a simple graph G that demonstrates that A is not stronger than B. We continue this line of research in the current paper.
As the main result, we show that for every graph A, there exists a simple graph D that covers A in such a way that the maximum number of pairwise disjoint perfect matchings equals the maximum number of pairwise disjoint perfect semi-matchings in A, i.e., spanning 1-regular subgraphs. Notably, the proof is constructive. As a corollary, we obtain a necessary condition for A to be stronger than B in general. This condition turns out to be sufficient whenever B is a 1-vertex graph (there are infinitely many of them), which, in particular, proves the aforementioned conjecture of Bok et al. in this case. Finally, we provide a constructive alternative to the existential NP-hardness proof of covering disconnected graphs in Bok et al. for the case when the target graph contains a 1-vertex component which itself determines an NP-hard covering problem.
Cite as
Filip Filipi, Jan Kratochvíl, and Roman Nedela. Generalized Snarks, Disjoint Perfect Matchings, and Graph Covers. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 45:1-45:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{filipi_et_al:LIPIcs.MFCS.2026.45,
author = {Filipi, Filip and Kratochv{\'\i}l, Jan and Nedela, Roman},
title = {{Generalized Snarks, Disjoint Perfect Matchings, and Graph Covers}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {45:1--45:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.45},
URN = {urn:nbn:de:0030-drops-274265},
doi = {10.4230/LIPIcs.MFCS.2026.45},
annote = {Keywords: graph, graph cover, perfect matching, NP-completeness}
}
Document
Generating Minimal Redundant and Maximal Irredundant Sets in Incidence Graphs
Abstract
It has been proved by Boros and Makino that there is no output-polynomial-time algorithm enumerating the minimal redundant sets or the maximal irredundant sets of a hypergraph, unless P = NP. The same question was left open for graphs, with only a few tractable cases known to date. In this paper, we focus on graph classes that capture incidence relations such as bipartite, co-bipartite, and split graphs, motivated by their strong relation with hypergraphs. Concerning maximal irredundant sets, we show that the problem on co-bipartite graphs is as hard as in general graphs and tractable in split and strongly orderable graphs, the latter being a generalization of chordal bipartite graphs. As for minimal redundant sets enumeration, we first show that the problem is intractable in split and co-bipartite graphs, answering the aforementioned open question. Then, we show that it is tractable on (C₃,C₅,C₆,C₈)-free graphs, a class of graphs incomparable to strongly orderable graphs, and which also generalizes chordal bipartite graphs. Our positive results rely on the structural properties of these graph classes and thus cannot be easily extended to bipartite graphs, for which the question remains open for both problems.
Cite as
Emanuel Castelo, Jérémie Chalopin, Oscar Defrain, and Simon Vilmin. Generating Minimal Redundant and Maximal Irredundant Sets in Incidence Graphs. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 46:1-46:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{castelo_et_al:LIPIcs.MFCS.2026.46,
author = {Castelo, Emanuel and Chalopin, J\'{e}r\'{e}mie and Defrain, Oscar and Vilmin, Simon},
title = {{Generating Minimal Redundant and Maximal Irredundant Sets in Incidence Graphs}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {46:1--46:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.46},
URN = {urn:nbn:de:0030-drops-274277},
doi = {10.4230/LIPIcs.MFCS.2026.46},
annote = {Keywords: Enumeration algorithms, maximal irredundant sets, minimal redundant sets, incidence graphs}
}
Document
Hamming Distance Between Finite Transducers
Abstract
We study bounded deviation of non-deterministic finite transducers under the Hamming distance: the bounded comparison problem asks, given two transducers and k ∈ ℕ, whether for every input the two transducers produce words at Hamming distance at most k. This problem is known to be decidable in polynomial time when k is fixed, and in co-NP otherwise.
We show that the problem is NL-complete when k is fixed, co-NP-complete when k is given in binary, and it is DP-complete to decide if the distance is exactly k. We also prove that if the two transducers have bounded comparison, then the maximal distance is at most quadratic in the size of both transducers, and that this bound is asymptotically tight.
We prove the results on deviation problems, which asks similar questions on the distance of the pairs of input and output of a single transducer, and show that these two families of problems are logspace many-one equivalent.
Cite as
Luc Dartois, Pierre-Cyrille Héam, Ismaël Jecker, and Silvio Vescovo. Hamming Distance Between Finite Transducers. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 47:1-47:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dartois_et_al:LIPIcs.MFCS.2026.47,
author = {Dartois, Luc and H\'{e}am, Pierre-Cyrille and Jecker, Isma\"{e}l and Vescovo, Silvio},
title = {{Hamming Distance Between Finite Transducers}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {47:1--47:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.47},
URN = {urn:nbn:de:0030-drops-274285},
doi = {10.4230/LIPIcs.MFCS.2026.47},
annote = {Keywords: Transducers, Hamming distance, NL-completeness, DP-completeness}
}
Document
Hardness of Approximation for Ground State Problems
Abstract
After nearly two decades of research, the question of a quantum PCP theorem for quantum Constraint Satisfaction Problems (CSPs) remains wide open. As a result, proving QMA-hardness of approximation for ground state energy estimation, analogous to hardness of approximation for MAX-k-CSP, has remained elusive. (QMA is Quantum Merlin-Arthur, a quantum generalization of NP with a quantum proof and quantum verifier.) Recently, it was shown [Bittel, Gharibian, Kliesch, CCC 2023] that a natural problem involving variational quantum circuits is QCMA-hard to approximate within ratio N^{1-ε} for any ε > 0 and N the input size. (Quantum Classical Merlin-Arthur is QMA, but with a classical proof.) Unfortunately, this problem was not related to quantum CSPs, leaving the question of hardness of approximation for quantum CSPs open.
In this work, we show that if instead of focusing on ground state energies (analogous to the optimal number of satisfied clauses), one considers computing properties of the ground space (analogous to computing properties of the MAX-k-CSP solution space), QCMA-hardness of computing ground space properties can be shown. In particular, we show that it is (1) QCMA-complete within ratio N^{1-ε} to approximate the Ground State Connectivity problem (GSCON), and (2) QCMA-hard within the same ratio to estimate the amount of entanglement of a local Hamiltonian’s ground state, denoted Ground State Entanglement (GSE). As a bonus, a simplification of our construction yields NP-completeness of approximation for a natural k-SAT reconfiguration problem, to be contrasted with the recent PCP-based PSPACE-hardness of approximation results for a different definition of k-SAT reconfiguration [Karthik C.S. and Manurangsi, 2023, and Hirahara, Ohsaka, STOC 2024].
Cite as
Sevag Gharibian and Carsten Hecht. Hardness of Approximation for Ground State Problems. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 48:1-48:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gharibian_et_al:LIPIcs.MFCS.2026.48,
author = {Gharibian, Sevag and Hecht, Carsten},
title = {{Hardness of Approximation for Ground State Problems}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {48:1--48:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.48},
URN = {urn:nbn:de:0030-drops-274299},
doi = {10.4230/LIPIcs.MFCS.2026.48},
annote = {Keywords: Quantum complexity, hardness of approximation, local Hamiltonians, ground state connectivity, reconfiguration}
}
Document
How Long Can The Escaping Ant Be Confined?
Abstract
Langton’s ant is a simple two-dimensional cellular automaton whose long-term behavior exhibits remarkable complexity. While it is known that the ant eventually escapes any finite connected region of the grid, the quantitative aspects of this escape remain poorly understood. In this paper, we study the escaping time of Langton’s ant, defined as the maximum number of steps the ant can perform within a finite connected domain before leaving it.
We establish general upper bounds on the escaping time as a function of the domain size, and derive improved bounds for rectangular domains. In particular, we obtain a factorial upper bound for square domains via an inductive decomposition argument. We also obtain linear upper bounds for rectangular domains of height two and three via a column-by-column analysis. More generally, for rectangular domains with a fixed height, we establish a polynomial upper bound in the number of columns. These results are complemented by exact values computed through an optimized simulation algorithm that exploits the geometric symmetries of the grid and employs a backtracking branching strategy to avoid exhaustive search over all color configurations. We also provide lower-bound constructions, proving that the linear upper bounds for rectangular domains of heights two and three are asymptotically optimal.
Cite as
Kossi Roland Etse. How Long Can The Escaping Ant Be Confined?. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 49:1-49:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{etse:LIPIcs.MFCS.2026.49,
author = {Etse, Kossi Roland},
title = {{How Long Can The Escaping Ant Be Confined?}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {49:1--49:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.49},
URN = {urn:nbn:de:0030-drops-274302},
doi = {10.4230/LIPIcs.MFCS.2026.49},
annote = {Keywords: Langton’s Ant, Escaping Time, Finite Grid Dynamics, Combinatorial Bounds, Discrete Dynamical Systems, Cellular Automata}
}
Document
Hypergraphs for Compact Closed Categories
Abstract
In recent years a succession of papers have taken to representing string diagrams as hypergraphs, the advantage of which is that the structural equations of diagrams come for free from the hypergraph structure. This improves implementability and reduces the number of rewrites necessary for reasoning.
Notably, however, string diagrams for compact closed categories have escaped this treatment. In this paper we introduce a combinatorial interpretation of compact closed string diagrams, via the Int construction on hypergraphs for traced string diagrams. Using this interpretation, we characterise string diagram rewriting as a suitable adaptation of double-pushout rewriting.
Cite as
Alessandro Di Giorgio, Alessandro Di Giorgio, Callum Reader, and Callum Reader. Hypergraphs for Compact Closed Categories. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 50:1-50:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{digiorgio_et_al:LIPIcs.MFCS.2026.50,
author = {Di Giorgio, Alessandro and Reader, Callum},
title = {{Hypergraphs for Compact Closed Categories}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {50:1--50:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.50},
URN = {urn:nbn:de:0030-drops-274318},
doi = {10.4230/LIPIcs.MFCS.2026.50},
annote = {Keywords: rewriting, compact closed categories, string diagrams}
}
Document
Improved Results for Knapsack with Removal
Abstract
We study the proportional online knapsack problem with removal. For randomized algorithms, we tighten the gap between the current lower and upper bounds on the expected competitive ratio by presenting a lower bound of roughly 1.27. We further study this problem under the model of online algorithms with predictions. Our lower bound arguments are agnostic to the type of available prediction, which makes them very general. For deterministic algorithms, we provide a tightly matching upper bound on the competitive ratio for a specific kind of weight prediction.
Cite as
Matthias Gehnen, Kübra Güven, Valentin Hächler, Dennis Komm, and Richard Královič. Improved Results for Knapsack with Removal. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 51:1-51:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gehnen_et_al:LIPIcs.MFCS.2026.51,
author = {Gehnen, Matthias and G\"{u}ven, K\"{u}bra and H\"{a}chler, Valentin and Komm, Dennis and Kr\'{a}lovi\v{c}, Richard},
title = {{Improved Results for Knapsack with Removal}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {51:1--51:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.51},
URN = {urn:nbn:de:0030-drops-274331},
doi = {10.4230/LIPIcs.MFCS.2026.51},
annote = {Keywords: Online computation, competitive analysis, knapsack problem, predictions}
}
Document
Increasing Arc-Connectivity by Bounded- and Fixed-Size Inversions
Abstract
Given an integer k ⩾ 1, a digraph D is k-arc-strong if the removal of any set of at most k-1 arcs of D yields a strongly connected digraph. For a digraph D and some set X ⊆ V(D), the inversion of X is the operation of flipping all arcs both of whose endvertices are in X. We initiate the study of establishing arc-connectivity properties by applying inversions of bounded or fixed size.
For fixed-size inversions, we consider the feasibility of the problem by characterizing, for all integers p ⩾ 2 and k ⩾ 1, the digraphs that can be made k-arc-strong by applying inversions of size exactly p, provided a minimum size of the digraphs.
For bounded-size inversions, the tractability of the feasibility problem follows easily from a famous theorem of Nash-Williams, so we focus on minimising the number of inversions. We prove that for all integers p ⩾ 3 and k ⩾ 1 and any ε > 0, there exists a polynomial-time (4k-2+ε)-approximation algorithm for computing the minimum number of inversions of size at most p that make a given digraph k-arc-strong. This is in stark contrast to other results on inversion optimization problems. On the other hand, we show that for any p ⩾ 3 and k ⩾ 1 the problem is NP-hard, and, moreover, APX-hard.
As a result on parameterized complexity, we show that for any k ⩾ 2, it is W[1]-hard with respect to p to decide whether a given digraph can be made k-arc-strong by applying a single inversion of size at most p. We also prove that for a given multidigraph, it is W[1]-hard with respect to 𝓁 to decide whether it can be made 2-arc-strong by applying 𝓁 inversions of size 2.
Cite as
Florian Hörsch and Lucas Picasarri-Arrieta. Increasing Arc-Connectivity by Bounded- and Fixed-Size Inversions. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 52:1-52:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{horsch_et_al:LIPIcs.MFCS.2026.52,
author = {H\"{o}rsch, Florian and Picasarri-Arrieta, Lucas},
title = {{Increasing Arc-Connectivity by Bounded- and Fixed-Size Inversions}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {52:1--52:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.52},
URN = {urn:nbn:de:0030-drops-274340},
doi = {10.4230/LIPIcs.MFCS.2026.52},
annote = {Keywords: Bounded-size Inversions, Strong Connectivity, Approximation Algorithms, Parameterized Complexity}
}
Document
Kernelization Bounds for Constrained Coloring
Abstract
We study the kernel complexity of constraint satisfaction problems over a finite domain, parameterized by the number of variables, whose constraint language consists of two relations: the non-equality relation and an additional permutation-invariant relation R. We establish a conditional lower bound on the kernel size in terms of the largest arity of an or relation definable from R. Building on this, we investigate the kernel complexity of uniformly rainbow free coloring problems. In these problems, for fixed positive integers d, 𝓁, and q ≥ d, we are given a graph G on n vertices and a collection F of 𝓁-tuples of d-subsets of its vertex set, and the goal is to decide whether there exists a proper coloring of G with q colors such that no 𝓁-tuple in F is uniformly rainbow, that is, no tuple has all its sets colored with the same d distinct colors. We determine, for all admissible values of d, 𝓁, and q, the infimum over all values η for which the problem admits a kernel of size O(n^η), under the assumption NP ⊈ coNP/poly. As applications, we obtain nearly tight bounds on the kernel complexity of various coloring problems under diverse settings and parameterizations. This includes graph coloring problems parameterized by the vertex-deletion distance to a disjoint union of cliques, resolving a question of Schalken (2020), as well as uniform hypergraph coloring problems parameterized by the number of vertices, extending results of Jansen and Pieterse (2019) and Beukers (2021).
Cite as
Ishay Haviv. Kernelization Bounds for Constrained Coloring. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 53:1-53:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{haviv:LIPIcs.MFCS.2026.53,
author = {Haviv, Ishay},
title = {{Kernelization Bounds for Constrained Coloring}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {53:1--53:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.53},
URN = {urn:nbn:de:0030-drops-274352},
doi = {10.4230/LIPIcs.MFCS.2026.53},
annote = {Keywords: parameterized complexity, kernelization, constraint satisfaction problems, coloring problems}
}
Document
Logics for Context-Free Hyperproperties
Abstract
We introduce a novel logic for the specification of context-free hyperproperties, which capture, e.g., the flow of information in security-critical recursive systems. Intuitively, the logic extends visibly pushdown automata by quantification over traces, just like HyperLTL, the most important logic for regular hyperproperties, extends LTL by quantification over traces. Using a game-based approach, we show that model-checking is decidable for formulas with a single quantifier alternation, provided the stack height of the visibly pushdown automaton only depends on the traces bound to the variables of the first quantifier block. A single quantifier alternation suffices to express many information-flow properties studied in the literature. Complementarily, we show that model-checking is undecidable for formulas with a single quantifier alternation, if the stack behavior of the visibly pushdown automaton may depend on the second quantifier block. This also implies that model-checking is undecidable for almost all fragments with more than one quantifier alternation.
Cite as
Sarah Winter and Martin Zimmermann. Logics for Context-Free Hyperproperties. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 54:1-54:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{winter_et_al:LIPIcs.MFCS.2026.54,
author = {Winter, Sarah and Zimmermann, Martin},
title = {{Logics for Context-Free Hyperproperties}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {54:1--54:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.54},
URN = {urn:nbn:de:0030-drops-274361},
doi = {10.4230/LIPIcs.MFCS.2026.54},
annote = {Keywords: Hyperproperties, model-checking, context-free languages}
}
Document
Lower Bounds for Meta-Reconfiguration
Abstract
In this paper, we explore the limits of algorithmic meta-theorems for combinatorial reconfiguration on graphs and prove several intractability results for highly restricted cases, which tightly complement the positive results by Mouawad et al. [IPEC 2014] and Gima et al. [Algorithmica 2024]. In this setting, we study reconfiguration problems on graphs in which the feasible sets are defined by formulas of first-order or monadic second-order logic: for a formula φ(X) with a free set variable X, the problem asks whether two given sets are connected by a token-jumping sequence in which every set satisfies φ on the input graph.
Our main contribution is to show that the problem is intractable even for first-order logic and for severely restricted graphs, such as paths and disjoint unions of stars or cliques. Combined with known results, these results settle the parameterized complexity for most of the well-studied structural parameters. We also study the setting where the sets to be reconfigured are small, i.e., their size is part of the parameter, and show that even in this setting the problem is hard for caterpillars, whereas it becomes tractable even for monadic second-order logic when parameterized additionally by shrub-depth.
Cite as
Kord Eickmeyer, Tatsuya Gima, Michael Lampis, Valia Mitsou, Edouard Nemery, Yota Otachi, Manolis Vasilakis, and Daniel Vaz. Lower Bounds for Meta-Reconfiguration. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 55:1-55:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{eickmeyer_et_al:LIPIcs.MFCS.2026.55,
author = {Eickmeyer, Kord and Gima, Tatsuya and Lampis, Michael and Mitsou, Valia and Nemery, Edouard and Otachi, Yota and Vasilakis, Manolis and Vaz, Daniel},
title = {{Lower Bounds for Meta-Reconfiguration}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {55:1--55:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.55},
URN = {urn:nbn:de:0030-drops-274373},
doi = {10.4230/LIPIcs.MFCS.2026.55},
annote = {Keywords: Combinatorial Reconfiguration, Token Jumping, Algorithmic Meta-Theorem, Fixed-Parameter Tractability}
}
Document
Maximum Matchings and Short Voting Paths
Abstract
Our input is a marriage instance G = (A ∪ B, E), i.e., it is a bipartite graph where every vertex has strict preferences over its neighbors. The preferences that a vertex has on its neighbors extend naturally to preferences over matchings. A maximum matching M that does not lose an election against any maximum matching (where vertices cast votes) is a popular maximum-matching. These matchings are useful in practice - they always exist and can be efficiently computed [Kavitha, SICOMP 2014]. Suppose preferences change; then the problem is to update the current matching M_0 via a short voting path to a popular maximum-matching, where a length-𝓁 voting path from M_0 to M_𝓁 is a sequence of maximum matchings ⟨M_0,M_1,…,M_𝓁⟩ such that each matching is more popular than its predecessor. There are maximum matchings from which there is no voting path (of any length) to a popular maximum-matching [Bhattacharya et al., ICALP 2015].
We show a polynomial-time algorithm to decide if there exists a short voting path, i.e. one of length ≤ 2, from a given maximum matching to a popular maximum-matching and find one, if so. Voting paths motivate natural relaxations of popularity: pseudo-popular maximum-matchings and mostly-popular maximum-matchings; these yield more egalitarian or optimal solutions than popular maximum-matchings. We show polynomial-time algorithms to compute such optimal solutions that go beyond popularity. In particular, we give a combinatorial characterization of pseudo-popular maximum-matchings in terms of forced vertices and forbidden edges. We also show a polynomial-time algorithm to compute at most |E| = m popular maximum-matchings such that any maximum matching that loses to some popular maximum-matching loses to at least one of these m matchings.
Cite as
Telikepalli Kavitha. Maximum Matchings and Short Voting Paths. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 56:1-56:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kavitha:LIPIcs.MFCS.2026.56,
author = {Kavitha, Telikepalli},
title = {{Maximum Matchings and Short Voting Paths}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {56:1--56:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.56},
URN = {urn:nbn:de:0030-drops-274380},
doi = {10.4230/LIPIcs.MFCS.2026.56},
annote = {Keywords: Bipartite graphs, Fractional matchings, Polytopes, LP duality}
}
Document
Model Checking with Temporal Graphs and Their Derivative
Abstract
Temporal graphs are graphs where the presence or properties of their vertices and edges change over time. When time is discrete, a temporal graph can be defined as a sequence of static graphs over a discrete time span, called lifetime, or as a single graph where each edge is associated with a specific set of time instants where the edge is alive. For static graphs, Courcelle’s Theorem asserts that any graph problem expressible in monadic second-order logic can be solved in linear time on graphs of bounded tree-width. We propose the first adaptation of Courcelle’s Theorem for monadic second-order logic on temporal graphs that does not explicitly rely on a parameter proportional to the lifetime, or defined as the maximum number of time-edges incident with any vertex which in the worst case is higher than the lifetime. We then introduce the notion of derivative over a sliding time window of a chosen size, and define the tree-width and twin-width of the temporal graph’s derivative. We exemplify its usefulness with meta-theorems with respect to a temporal variant of first-order logic. The resulting logic expresses a wide range of temporal graph problems including a version of temporal cliques, an important notion when querying time series databases for community structures.
Cite as
Binh-Minh Bui-Xuan, Florent Krasnopol, Bruno Monasson, and Nathalie Sznajder. Model Checking with Temporal Graphs and Their Derivative. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 57:1-57:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{buixuan_et_al:LIPIcs.MFCS.2026.57,
author = {Bui-Xuan, Binh-Minh and Krasnopol, Florent and Monasson, Bruno and Sznajder, Nathalie},
title = {{Model Checking with Temporal Graphs and Their Derivative}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {57:1--57:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.57},
URN = {urn:nbn:de:0030-drops-274393},
doi = {10.4230/LIPIcs.MFCS.2026.57},
annote = {Keywords: temporal graphs, dynamic network, tree decomposition, monadic second order logic, first order logic, derivative}
}
Document
Monotone Bounded Depth Formula Complexity of Graph Homomorphism Polynomials
Abstract
We introduce baggy elimination trees, a novel graph decomposition that generalises the classical elimination trees underlying treedepth, and use them to give a complete characterisation of the monotone bounded-depth formula complexity of graph homomorphism and coloured isomorphism polynomials. Specifically, we prove that the Δ-product depth monotone formula complexity of these polynomials is Θ(n^λ_Δ(H)), where λ_Δ(H) is the minimum cost of a baggy elimination tree for H at BET-depth Δ.
This result closes the last open case in the programme initiated by Komarath, Pandey and Rahul [Balagopal Komarath et al., 2023] and continued by Bhargav, Chen, Curticapean and Dwivedi [C. S. Bhargav et al., 2025]: tight size characterisations of monotone circuit complexity (via treewidth / bounded-depth treewidth), monotone ABP complexity (via pathwidth / bounded-depth pathwidth), and monotone formula complexity (via treedepth) were already known; our theorem supplies the missing bounded-depth formula characterisation via the new notion of bounded-depth baggy-elimination-tree cost λ_Δ, completing the picture for all three models in algebraic complexity and their fixed depth variants.
As applications, for constant-degree polynomial families we derive an almost-optimal separation between monotone circuits and monotone formulas at every fixed product depth: there exists a family computable by O(N)-size monotone circuits of product depth Δ that requires Ω(N^{Δ/2})-size monotone formulas of the same depth (and this exponent is optimal up to a constant factor). We also prove a strict depth hierarchy: for every Δ ≥ 1 and every constant k ≥ 2, there is a constant-degree family with O(s(N))-size monotone formulas of product depth Δ that requires Ω(s(N)^k)-size monotone formulas of product depth Δ - 1.
Cite as
Balagopal Komarath and Rohit Narayanan. Monotone Bounded Depth Formula Complexity of Graph Homomorphism Polynomials. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 58:1-58:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{komarath_et_al:LIPIcs.MFCS.2026.58,
author = {Komarath, Balagopal and Narayanan, Rohit},
title = {{Monotone Bounded Depth Formula Complexity of Graph Homomorphism Polynomials}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {58:1--58:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.58},
URN = {urn:nbn:de:0030-drops-274406},
doi = {10.4230/LIPIcs.MFCS.2026.58},
annote = {Keywords: Monotone complexity, bounded depth, formula complexity, graph homomorphism, algebraic complexity}
}
Document
Multi-Head Finite-State Dimension
Abstract
We introduce multi-head finite-state dimension, a generalization of finite-state dimension in which a group of finite-state agents (the heads) with oblivious, one-way movement rules, each reporting only one symbol at a time, enable their leader to bet on subsequent symbols in an infinite data stream. In aggregate, such a scheme constitutes an h-head finite state gambler whose maximum achievable growth rate of capital in this task, quantified using betting strategies called gales, determines the multi-head finite-state dimension of the sequence. The 1-head case is equivalent to finite-state dimension as defined by Dai, Lathrop, Lutz and Mayordomo (2004). In our main theorem, we prove a strict hierarchy as the number of heads increases, giving an explicit sequence family that separates, for each positive integer h, the earning power of h-head finite-state gamblers from that of (h+1)-head finite-state gamblers. We prove that multi-head finite-state dimension is stable under finite unions but that the corresponding quantity for any fixed number h > 1 of heads - the h-head finite-state predimension - lacks this stability property.
Cite as
Xiang Huang, Xiaoyuan Li, Jack H. Lutz, and Neil Lutz. Multi-Head Finite-State Dimension. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 59:1-59:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{huang_et_al:LIPIcs.MFCS.2026.59,
author = {Huang, Xiang and Li, Xiaoyuan and Lutz, Jack H. and Lutz, Neil},
title = {{Multi-Head Finite-State Dimension}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {59:1--59:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.59},
URN = {urn:nbn:de:0030-drops-274410},
doi = {10.4230/LIPIcs.MFCS.2026.59},
annote = {Keywords: Finite-state dimension, effective dimension, algorithmic randomness}
}
Document
Multi-Prover Interactive Proof Systems with Leakage
Abstract
It is known that there exist multi-prover interactive protocols (MIP protocols) for the complexity class NEXP, succinct MIP protocols for NP and multi-prover interactive protocols with shared entanglement (MIP^∗ protocols) for RE. This extraordinary power of multi-prover interactive proof systems comes from the assumption that provers do not communicate with each other during the protocols. If they are allowed to communicate freely, the setting is the same as in the single-prover case, and the computational power of the system becomes significantly weaker.
In this paper, we investigate for the first time the setting where communication (i.e., leakage of information) between provers is allowed but bounded. We introduce two techniques to approach this question and show that multi-prover interactive proof systems are robust against some amount of leakage. Our first technique is based on parallel repetition theorems. We apply it to show that for any polynomial p, we can construct two-prover one-round MIP and MIP^∗ protocols for NEXP and RE, respectively, that are robust against p(n) bits of leakage. We further derive our second technique to convert any low-soundness PCP construction to a two-prover one-round MIP protocol for NP robust against leakage. We also discuss the relation between robustness against leakage in multi-prover interactive proof systems and the Sliding Scale Conjecture in the PCP literature.
Cite as
Vahid R. Asadi, Atsuya Hasegawa, and François Le Gall. Multi-Prover Interactive Proof Systems with Leakage. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 60:1-60:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{asadi_et_al:LIPIcs.MFCS.2026.60,
author = {Asadi, Vahid R. and Hasegawa, Atsuya and Le Gall, Fran\c{c}ois},
title = {{Multi-Prover Interactive Proof Systems with Leakage}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {60:1--60:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.60},
URN = {urn:nbn:de:0030-drops-274425},
doi = {10.4230/LIPIcs.MFCS.2026.60},
annote = {Keywords: Multi-prover interactive proof systems}
}
Document
Nearly Tight Bounds on the Block Number of Boolean Functions in Terms of Sensitivity
Abstract
This paper explores the previously studied measure called block number of Boolean functions, that counts the maximum possible number of minimal sensitive blocks for any input. We present close to tight upper bounds on the block number in terms of the function’s sensitivity and the allowed block size, improving previous bounds by a quadratic factor. Moreover, our bound on the block number yields sharper upper bounds on DNF size and decision tree size. For some functions, our upper bounds on decision tree size and DNF size are exponentially smaller than those obtained by previous methods. We obtain these results by introducing and estimating a novel measure called brick number, which not only upper bounds the block number but also leads to a new characterization of block sensitivity.
Cite as
Sourav Chakraborty and Anna Gál. Nearly Tight Bounds on the Block Number of Boolean Functions in Terms of Sensitivity. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 61:1-61:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chakraborty_et_al:LIPIcs.MFCS.2026.61,
author = {Chakraborty, Sourav and G\'{a}l, Anna},
title = {{Nearly Tight Bounds on the Block Number of Boolean Functions in Terms of Sensitivity}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {61:1--61:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.61},
URN = {urn:nbn:de:0030-drops-274432},
doi = {10.4230/LIPIcs.MFCS.2026.61},
annote = {Keywords: Boolean functions, sensitivity, block sensitivity, block number}
}
Document
On CC⁰ Lower Bounds for AND via Torus Polynomials
Abstract
We explore the torus polynomial approximation based approach towards a long-standing question: whether AND can be computed by CC⁰ circuits - the class of constant-depth polynomial size circuits containing MOD_m gates for some natural number m. Bhrushundi, Hosseini, Lovett and Rao (ITCS 2019) introduced torus polynomial approximations as an approach for proving lower bounds against ACC⁰ - a class containing CC⁰ where the circuits are also allowed AND, OR and NOT gates.
We show how lower bounds for torus polynomials approximating AND can be used to make progress on this question. Using lower bounds on the degree of symmetric torus polynomials approximating AND, proved by Krishan and Vishwanathan (ITCS 2026), we prove size lower bounds for symmetric CC⁰-circuits computing AND. More precisely, we prove that any depth h symmetric CC⁰ circuit requires 2^Ω̃(n^{1/O(h)}) size to compute AND.
A key ingredient in our proof is an argument that we can construct symmetric torus polynomials to approximate symmetric CC⁰ circuits. Our construction exhibits an explicit correspondence between the symmetry of the circuit and that of the polynomial. Using this, we also establish lower bounds for weaker notions of circuit symmetry. Lower bounds for symmetric CC⁰ circuits were also independently established by Pago (ICALP 2026) using different techniques.
In the asymmetric regime, we establish degree upper bounds for depth three circuits of the form MOD_p∘MOD_m∘AND_O(1) where m = pq is a semiprime. This circuit class is a special case of the constant degree hypothesis, introduced by Barrington, Straubing and Thérien (Information and Computation, 1990), where m could be an arbitrary composite number. We argue that improved lower bounds for asymmetric torus polynomials approximating AND imply size lower bounds for semiprime m and hence progress on the constant-degree hypothesis.
Cite as
Vaibhav Krishan and Jayalal Sarma. On CC⁰ Lower Bounds for AND via Torus Polynomials. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 62:1-62:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{krishan_et_al:LIPIcs.MFCS.2026.62,
author = {Krishan, Vaibhav and Sarma, Jayalal},
title = {{On CC⁰ Lower Bounds for AND via Torus Polynomials}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {62:1--62:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.62},
URN = {urn:nbn:de:0030-drops-274449},
doi = {10.4230/LIPIcs.MFCS.2026.62},
annote = {Keywords: Circuit complexity, CC⁰, constant degree hypothesis, torus polynomials}
}
Document
On Equivalent Characterizations of the Polynomial Hierarchy in Abstract Models of Computation
Abstract
We investigate machine models similar to Turing machines that are augmented with the operations of a first-order structure ℛ, and we show that under weak conditions on ℛ, the complexity class Σ_kℛ may be characterized in four equivalent ways: (1) by polynomial-time algorithms implemented on ℛ-machines together with witness strings, (2) by the Σ_k ℛ-complete problem Σ_k SAT(ℛ), (3) by k-th existential fragment second-order metafinite logic over ℛ via descriptive complexity, and (4) via oracles. By characterizing Σ_k ℛ in these four ways, we extend previous work and embed it in one coherent framework. In addition, we derive similar results for ∃_k ℛ, the constant-free Boolean part of Σ_k ℛ, by showing that ∃_k ℛ may be characterized in four analogous ways.
Some conditions on ℛ must be assumed in order to achieve the above quaternity because there are infinite-vocabulary structures for which NP(ℛ) = Σ₁ ℛ does not have a complete problem. Surprisingly, even in these cases, we show that NP(ℛ) does have a characterization in terms of existential second-order metafinite logic, suggesting that descriptive complexity theory is well suited to working with infinite-vocabulary structures, such as real vector spaces.
Cite as
Jeremy C. Kirn, Lucas Meijer, Tillmann Miltzow, and Hans L. Bodlaender. On Equivalent Characterizations of the Polynomial Hierarchy in Abstract Models of Computation. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 63:1-63:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kirn_et_al:LIPIcs.MFCS.2026.63,
author = {Kirn, Jeremy C. and Meijer, Lucas and Miltzow, Tillmann and Bodlaender, Hans L.},
title = {{On Equivalent Characterizations of the Polynomial Hierarchy in Abstract Models of Computation}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {63:1--63:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.63},
URN = {urn:nbn:de:0030-drops-274450},
doi = {10.4230/LIPIcs.MFCS.2026.63},
annote = {Keywords: Machines over a first-order structure, BSS machines, Cook Levin, Fagin, NP, existential theory of the reals, polynomial hierarchy, metafinite model theory, descriptive complexity, oracles}
}
Document
On Jumps, Interactions, and Intersection Types
Abstract
The Jumping Abstract Machine (JAM), an evaluation mechanism for the λ-calculus, was introduced by Danos and Regnier as an optimization of the Interaction Abstract Machine (IAM), itself an operational counterpart to Girard’s Geometry of Interaction and Abramsky et al. game semantics. Moreover, the JAM is isomorphic to the Pointer Abstract Machine (PAM), the syntactical counterpart of Hyland and Ong’s game semantics. We study a generalization of the JAM, that we call the Parametric Jumping Abstract Machine (PaJAM) and show that there is a tight correspondence between the PaJAM and non-idempotent intersection types: given a normalizing term t, the number of steps taken by the PaJAM when evaluating t can be extracted from its non-idempotent intersection type derivation. Remarkably, fixing the backtracking depth of the PaJAM, one can easily recover both the JAM/PAM, when the depth is constrained to be zero, and the IAM, when it is instead unconstrained. Exploiting type-theoretic machinery, we analyze the complexity of the PaJAM, showing that it is polynomial in the number of weak head β steps, giving rise to a reasonable cost model, for each finite bound on the backtracking depth.
Cite as
Stefano Catozi, Ugo Dal Lago, and Gabriele Vanoni. On Jumps, Interactions, and Intersection Types. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 64:1-64:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{catozi_et_al:LIPIcs.MFCS.2026.64,
author = {Catozi, Stefano and Dal Lago, Ugo and Vanoni, Gabriele},
title = {{On Jumps, Interactions, and Intersection Types}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {64:1--64:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.64},
URN = {urn:nbn:de:0030-drops-274468},
doi = {10.4230/LIPIcs.MFCS.2026.64},
annote = {Keywords: lambda-calculus, geometry of interaction, intersection types, abstract machines}
}
Document
On Positivity of Exponential-Trigonometric Polynomials and Irrationality Exponents
Abstract
We establish Diophantine hardness results for the decidability of the Positivity Problem for exponential-trigonometric polynomials over computable discrete subfields of the real numbers, and for related questions. We show that any algorithm for deciding either non-negativity, eventual non-negativity, the existence of a zero, or the existence of infinitely many zeros of exponential-trigonometric polynomials over a computable discrete subfield K of the reals containing the number π can be translated into an algorithm for computing the irrationality exponents of all elements of K. As a consequence, we exhibit a computable discrete subfield K of the reals such that all of the aforementioned questions about exponential-trigonometric polynomials over K are undecidable. In particular, we provide the first example of a natural generalisation of the Continuous Skolem Problem that is provably undecidable.
Cite as
Pieter Collins, Bernard Hanzon, and Eike Neumann. On Positivity of Exponential-Trigonometric Polynomials and Irrationality Exponents. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 65:1-65:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{collins_et_al:LIPIcs.MFCS.2026.65,
author = {Collins, Pieter and Hanzon, Bernard and Neumann, Eike},
title = {{On Positivity of Exponential-Trigonometric Polynomials and Irrationality Exponents}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {65:1--65:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.65},
URN = {urn:nbn:de:0030-drops-274478},
doi = {10.4230/LIPIcs.MFCS.2026.65},
annote = {Keywords: Linear Dynamical Systems, Computability, Computable Numbers, Transcendental Numbers, Irrationality Measure, Irrationality Exponent}
}
Document
On the Complexity of Locally Dense Lattices
Abstract
Locally dense lattices are central gadgets used to prove the hardness of the Shortest Vector Problem and related lattice problems. Informally, a locally dense lattice is a lattice ℒ that contains exponentially many lattice vectors
inside some 𝓁_p ball centered at 𝐬 with radius at most an α < 1 fraction of the length of its shortest nonzero lattice vector.
In this paper, taking a "meta" viewpoint on locally dense lattices, we introduce the Locally Dense Lattice Problem (LDLP), the decision problem of determining whether a given input specifies a locally dense lattice. Our main result is that LDLP in 𝓁_p norms for all finite p ≥ log₂ 3 and for the infinity norm is complete for the second level of the polynomial hierarchy.
We also compare two standard definitions of local density that appear in prior work. Micciancio’s original definition (FOCS 1998 and SICOMP 2001) uses integer coefficient vectors, while later work by Micciancio (ToC 2012) and by Bennett and Peikert (RANDOM 2023) uses short vectors in a shifted coset. We show that the corresponding promise problems are mutually reducible in deterministic polynomial time, which shows that the two formulations are robust.
Cite as
Shuichi Hirahara and Kazuki Ogitsuka. On the Complexity of Locally Dense Lattices. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 66:1-66:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hirahara_et_al:LIPIcs.MFCS.2026.66,
author = {Hirahara, Shuichi and Ogitsuka, Kazuki},
title = {{On the Complexity of Locally Dense Lattices}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {66:1--66:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.66},
URN = {urn:nbn:de:0030-drops-274485},
doi = {10.4230/LIPIcs.MFCS.2026.66},
annote = {Keywords: Lattice problems, Locally dense lattices}
}
Document
On the Parameterized Complexity of Bounded-Density Vertex Deletion
Abstract
We explore the parameterized complexity of Bounded Density Vertex Deletion (BDVD): given a graph G, an integer budget k, and a target density τ_ρ, the task is to determine whether the density (i.e. number of edges divided by number of vertices) of the densest subgraph of G can be reduced to at most τ_ρ by deleting at most k vertices. Our primary focus is on structural graph parameters related to treewidth, as the parameterized complexity of BDVD with respect to treewidth was left as open question by Bazgan et al. [JCSS, 2025]. We resolve this question by showing W[1]-hardness with respect to various parameters, including treedepth and feedback vertex number. These results imply W[1]-hardness with respect to treewidth. We obtain positive results for parameters larger than treedepth and feedback vertex number, namely we show BDVD is in FPT parameterized by the max leaf number or vertex integrity. Under the assumption that the target density τ_ρ is a fixed constant the parameterized complexity landscape of BDVD changes drastically, allowing a fixed-parameter tractable algorithm even for parameters smaller than treewidth, namely cliquewidth. Altogether, our results provide a refined complexity landscape for Bounded Density Vertex Deletion, sharply distinguishing between tractable and intractable parameter regimes under structural parameterizations.
Cite as
Jakob Raupach, Tom-Lukas Breitkopf, Anton Herrmann, and André Nichterlein. On the Parameterized Complexity of Bounded-Density Vertex Deletion. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 67:1-67:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{raupach_et_al:LIPIcs.MFCS.2026.67,
author = {Raupach, Jakob and Breitkopf, Tom-Lukas and Herrmann, Anton and Nichterlein, Andr\'{e}},
title = {{On the Parameterized Complexity of Bounded-Density Vertex Deletion}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {67:1--67:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.67},
URN = {urn:nbn:de:0030-drops-274497},
doi = {10.4230/LIPIcs.MFCS.2026.67},
annote = {Keywords: Graph Modification Problem, Integer Linear Programming, Dynamic Programming, Max Leaf Number, Vertex Integrity}
}
Document
On the Size Complexity of Two-Way Finite Automata with Drop-Once Pebbles
Abstract
A two-way finite automaton with drop-once pebbles (O. Martynova, A. Okhotin, "A time to cast away stones: On a family of pebble automata", IJFCS, 37 (2026)) may drop its pebbles at any squares of the tape, but a pebble once dropped cannot be moved anymore. In this paper, it is proved that transforming an n-state deterministic automaton with k drop-once pebbles to a standard two-way deterministic finite automaton (2DFA) requires Θ(n^{k+1}) states in the worst case. For nondeterministic two-way automata with one drop-once pebble, it is proved that transforming them to a 2DFA requires at least 2^{n/3-o(n)} states, transforming to a two-way nondeterministic automaton (2NFA) takes at least 2^{n/6-o(n)} states, and, finally, determinizing them to a deterministic two-way automaton with one drop-once pebble requires at least 2^{n/6-o(n)} states.
Cite as
Georgy Kipriyanov and Alexander Okhotin. On the Size Complexity of Two-Way Finite Automata with Drop-Once Pebbles. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 68:1-68:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kipriyanov_et_al:LIPIcs.MFCS.2026.68,
author = {Kipriyanov, Georgy and Okhotin, Alexander},
title = {{On the Size Complexity of Two-Way Finite Automata with Drop-Once Pebbles}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {68:1--68:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.68},
URN = {urn:nbn:de:0030-drops-274500},
doi = {10.4230/LIPIcs.MFCS.2026.68},
annote = {Keywords: Finite automata, two-way automata, pebble automata, determinization}
}
Document
On the Tension Between Full-Rankness and Self-Reducibility for Set-Multilinear Polynomials
Abstract
In this paper, we rule out a natural programme for proving VF ≠ VNP via set-multilinear formula lower bounds. The programme combines two ingredients present in the literature: the n^Ω(log n) set-multilinear formula lower bound of Kush and Saraf (CCC 2022) for any full-rank polynomial, and an IMM-style self-reducibility that propagates such a bound to small degree, where Raz’s set-multilinearisation (J. ACM 2013) converts it to a general formula lower bound. Each ingredient has been realised separately, yet no polynomial family is known to combine them. We prove that no such family can exist: under any polynomial-width IMM-style self-reducibility - i.e., a small-width expression g = ∑_{k=1}^w L_k ⋅ R_k with each summand factoring across a balanced split - full-rankness forces width n^Ω(d), and even approximate full-rankness across a near-balanced split forces width n^Ω(√d). This rules out the full-rank/self-reducible route to VF ≠ VNP.
Cite as
Deepanshu Kush. On the Tension Between Full-Rankness and Self-Reducibility for Set-Multilinear Polynomials. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 69:1-69:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kush:LIPIcs.MFCS.2026.69,
author = {Kush, Deepanshu},
title = {{On the Tension Between Full-Rankness and Self-Reducibility for Set-Multilinear Polynomials}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {69:1--69:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.69},
URN = {urn:nbn:de:0030-drops-274512},
doi = {10.4230/LIPIcs.MFCS.2026.69},
annote = {Keywords: Algebraic formula lower bounds, set-multilinear formulas, iterated matrix multiplication, rank methods, hardness escalation, self-reducibility, barriers}
}
Document
Online Firefighting on Cactus Graphs
Abstract
The firefighting game is a fundamental problem in theoretical computer science, modeling the containment of a spreading process under limited defensive resources. In each round, an algorithm may protect a set of vertices by placing firefighters on them, preventing the fire from spreading to those vertices. Vertices that are never reached by the fire are said to be saved. The goal is to maximize the number of saved vertices. While the offline version has been extensively studied, much less is known about the competitive complexity of the online variant, where the underlying graph is known in advance but the number of available firefighters in each round is revealed online.
We study how graph structure governs the power of the adversary in the online firefighting game. On trees, the problem is known to be 2-competitive (Coupechoux et al., 2019), a result that relies on a strong structural alignment between the online algorithm and the offline optimal solution throughout the process. We show that this alignment breaks down as soon as cycles are present. In particular, the algorithm and the offline optimal solution may break a cycle differently, or even break different cycles, and therefore operate on fundamentally different residual graphs. As a result, the adversary gains additional leverage by steering the process along residual graph structures that no longer admit a direct comparison between the algorithm and the offline optimal solution.
Our main result shows that the presence of a single cycle already increases the competitive complexity of the firefighting game dramatically. We first show that even on a tadpole graph (a cycle with a tail), no deterministic online algorithm can achieve a competitive ratio better than Ω(√n), where n is the number of vertices. We complement this lower bound with matching upper bounds by designing an O(√n)-competitive online algorithm for 1-almost trees, that is, graphs obtained from a tree by adding at most one edge. Furthermore, we extend our framework to cactus graphs and prove that, despite the presence of multiple cycles, the competitive complexity remains Θ(√n) as long as the cycles do not share edges. Together, these results yield a tight characterization of the adversarial power induced by cycles with non-overlapping edges. Finally, considering that cactus graphs have treewidth of 2, we study a variant in which firefighters are released in pairs, that is, an even number of firefighters becomes available in each round. Surprisingly, the competitive complexity is significantly reduced to 3-competitive for this setting.
Cite as
Max Hugen, Bob Krekelberg, and Alison Hsiang-Hsuan Liu. Online Firefighting on Cactus Graphs. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 70:1-70:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hugen_et_al:LIPIcs.MFCS.2026.70,
author = {Hugen, Max and Krekelberg, Bob and Liu, Alison Hsiang-Hsuan},
title = {{Online Firefighting on Cactus Graphs}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {70:1--70:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.70},
URN = {urn:nbn:de:0030-drops-274520},
doi = {10.4230/LIPIcs.MFCS.2026.70},
annote = {Keywords: Firefighting game, Online algorithms, Cactus graphs, 1-almost trees}
}
Document
Order-Invariant Cluster First-Order Logic on Graph Classes of Bounded Degree
Abstract
We introduce a new logic, called cluster first-order logic, a restricted fragment of first-order logic specifically designed to study order invariance. An order-invariant formula is one on a vocabulary that contains an order; however, whether a structure satisfies it or not is independent of the interpretation of the order.
We show that while order-invariant cluster first-order logic can define properties outside the scope of plain first-order logic in general, its expressive power is included in that of first-order logic when it comes to classes of bounded degree.
We establish this result by explicitly constructing linear orders such that similar structures remain similar when they are expanded with these orders. This similarity-preserving, local-to-global approach is technically involved and somewhat counterintuitive, since adding an order usually reveals distinctions that are otherwise hidden due to the locality of first-order logic. We believe that this work can be a stepping stone toward applying such techniques to plain first-order logic and toward settling the question of the expressive power of order-invariant plain first-order logic.
Cite as
Fatemeh Ghasemi and Julien Grange. Order-Invariant Cluster First-Order Logic on Graph Classes of Bounded Degree. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 71:1-71:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ghasemi_et_al:LIPIcs.MFCS.2026.71,
author = {Ghasemi, Fatemeh and Grange, Julien},
title = {{Order-Invariant Cluster First-Order Logic on Graph Classes of Bounded Degree}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {71:1--71:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.71},
URN = {urn:nbn:de:0030-drops-274533},
doi = {10.4230/LIPIcs.MFCS.2026.71},
annote = {Keywords: Order invariance, cluster first-order logic, model checking}
}
Document
Parameterized Complexity of Efficient Sortation
Abstract
A crucial challenge arising in the design of large-scale logistical networks is to optimize parcel sortation for routing. We study this problem under the recent graph-theoretic formalization of Van Dyk, Klause, Koenemann and Megow (IPCO 2024). The problem asks - given an input digraph D (the fulfillment network) together with a set of commodities represented as source-sink tuples - for a minimum-outdegree subgraph H of the transitive closure of D that contains a source-sink route for each of the commodities. Given the underlying motivation, we study two variants of the problem which differ in whether the routes for the commodities are fixed or can be chosen arbitrarily.
We perform a thorough parameterized analysis of the complexity of both problems, concentrating on three fundamental parameterizations:
1) When considering the target outdegree of H, we show that the problems are paraNP-hard even in highly restricted cases;
2) When parameterizing by the number of commodities, we utilize Ramsey-type arguments and the color-coding technique to obtain fixed-parameter algorithms for both problems;
3) When parameterizing by the structure of D, we establish fixed-parameter tractability for both problems w.r.t. the combined parameterization of treewidth, maximum degree and the maximum routing length. We complement this with lower bounds which show that omitting any of the three parameters results in paraNP-hardness.
Cite as
Robert Ganian, Hung P. Hoang, and Simon Wietheger. Parameterized Complexity of Efficient Sortation. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 72:1-72:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ganian_et_al:LIPIcs.MFCS.2026.72,
author = {Ganian, Robert and Hoang, Hung P. and Wietheger, Simon},
title = {{Parameterized Complexity of Efficient Sortation}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {72:1--72:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.72},
URN = {urn:nbn:de:0030-drops-274548},
doi = {10.4230/LIPIcs.MFCS.2026.72},
annote = {Keywords: sort point problem, parameterized complexity, treewidth}
}
Document
Parameterizing the Complexity of Finding Long Paths in DAGs
Abstract
Given a graph G and two vertices s and t, the Long Path problem asks whether there exists a path of length at least k from s to t. For general graphs, this problem is NP-hard when k is part of the input. For directed acyclic graphs (DAGs), however, it is solvable in polynomial time and is NL-complete.
A nondeterministic logspace computation is said to be unambiguous if, on every input, there is at most one accepting computation path. The class UL consists of all problems solvable by such machines, and whether NL = UL is an open question.
In this work, we study the unambiguous complexity of the Long Path problem on DAGs under parameterization. Specifically, we consider the problem of deciding whether there exists a path of length at least n-k between two given vertices in a DAG. Bhadra and Tewari [Ronak Bhadra and Raghunath Tewari, 2025] showed that this problem can be solved in unambiguous and co-unambiguous O(klog n) space. We improve this result by giving an algorithm that runs in unambiguous O(k+log n) space. Additionally, we obtain an algorithm that achieves unambiguous and co-unambiguous O(klog n) space while running in time polynomial in both n and k, improving the previous O^*(n^k) time bound.
Cite as
Ronak Bhadra, Saurya Singh, and Raghunath Tewari. Parameterizing the Complexity of Finding Long Paths in DAGs. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 73:1-73:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bhadra_et_al:LIPIcs.MFCS.2026.73,
author = {Bhadra, Ronak and Singh, Saurya and Tewari, Raghunath},
title = {{Parameterizing the Complexity of Finding Long Paths in DAGs}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {73:1--73:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.73},
URN = {urn:nbn:de:0030-drops-274557},
doi = {10.4230/LIPIcs.MFCS.2026.73},
annote = {Keywords: Unambiguous Computations, Directed Acyclic Graphs, Space Complexity}
}
Document
Primitive Recursion Without Composition
Abstract
What computational mechanisms do recurrent neural networks, polynomial ordinary differential equations, and discrete polynomial maps each bring to the table, and what do they lack? All three are models of computation over the continuum: they operate on real-valued states and evolve by real-valued dynamics, even when the functions we ask them to compute are ultimately discrete. We investigate how these models compare, their strengths, their limitations, and the precise resources on which each one relies, through the lens of primitive recursive functions.
We prove that the classical notion of primitive recursion admits equivalent characterizations in all three dynamical frameworks: bounded iteration of a fixed recurrent ReLU network, robust computation by a fixed polynomial ordinary differential equation, and iteration of a fixed polynomial map in discrete time with an externally supplied step-size parameter. In each case, the time bound is itself primitive recursive, composition is not postulated as a closure rule but emerges from the dynamics, and the input is given as a raw integer vector with no auxiliary encoding. At the proof level, every primitive recursive function is first compiled into bounded iteration of a single threshold-affine normal form map, which is then interpreted as a recurrent ReLU computation on the one hand, and as a robust polynomial ODE on the other.
The equivalences expose a structural asymmetry between discrete and continuous polynomial computation. We prove that no fixed polynomial map can round uniformly toward the nearest integer, and that none can realize exact phase selection: two operations that polynomial ODEs perform robustly through their continuous-time flow. Each formalism compensates for a limitation that the others do not share: the ReLU gate provides exact branching, continuous time provides autonomous rounding and control, and the step-size parameter recovers both at the cost of discretization precision. Our equivalence theorem characterizes what each resource contributes, and opens the way to dynamical characterizations of subrecursive hierarchies and complexity classes by restricting the time bounds, polynomial degrees, or discretization resources within the same framework.
More broadly, the constructions reveal that these real-valued models do not compute by composing subroutines in the classical sense: they compute by shaping the trajectory of a dynamical system, through clocks, phase selectors, stabilization mechanisms, and error correction built into the dynamics itself. This is a mode of computation that differs structurally from symbolic programming, and our equivalence theorem provides a precise framework in which the difference can be studied.
Cite as
Olivier Bournez. Primitive Recursion Without Composition. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 74:1-74:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bournez:LIPIcs.MFCS.2026.74,
author = {Bournez, Olivier},
title = {{Primitive Recursion Without Composition}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {74:1--74:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.74},
URN = {urn:nbn:de:0030-drops-274564},
doi = {10.4230/LIPIcs.MFCS.2026.74},
annote = {Keywords: Discrete ordinary differential equations, Finite Differences, Implicit complexity, Recursion scheme, Ordinary differential equations, Models of computation, Analog Computations, Formal neural networks}
}
Document
Product-State Approximation Algorithms for the Transverse Field Ising Model
Abstract
We study classical polynomial-time approximation algorithms for the transverse field Ising model (TFIM), allowing a mixture of ferromagnetic and antiferromagnetic interactions between pairs of qubits, alongside transverse field terms with arbitrary non-negative weights.
In this work, we first prove a second-order conic inequality based on the anticommutation property of the two competing terms (Ising Z_i Z_j vs. field X_i terms), and we use this inequality to strengthen the basic SDP relaxation of the problem. By producing two competing rounded product state solutions and taking the better of the two we achieve an approximation ratio γ≈ 0.7860. A further improvement by non-uniform interpolation achieves a ratio γ ≈ 0.82197. Finally, we give an explicit purely antiferromagnetic TFIM instance on three qubits for which every product state achieves at most 169/180≈ 0.9389 of the true optimum, yielding an upper bound for all algorithms producing product state approximations, even in the purely antiferromagnetic case.
Cite as
Vincenzo Lipardi, David Mestel, and Georgios Stamoulis. Product-State Approximation Algorithms for the Transverse Field Ising Model. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 75:1-75:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{lipardi_et_al:LIPIcs.MFCS.2026.75,
author = {Lipardi, Vincenzo and Mestel, David and Stamoulis, Georgios},
title = {{Product-State Approximation Algorithms for the Transverse Field Ising Model}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {75:1--75:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.75},
URN = {urn:nbn:de:0030-drops-274574},
doi = {10.4230/LIPIcs.MFCS.2026.75},
annote = {Keywords: Ising model, Hamiltonian Complexity, Approximation Algorithms, Semidefinite Programming}
}
Document
Quantitative Equational Rewriting
Abstract
Rewriting logic is a logical framework for expressing both concurrent computation and logical deduction using equations and rewrite rules. Quantitative equational reasoning enriches equations with quantitative measures, expressing concepts such as similarity or proximity rather than mere equality of terms. In this article, we bring these two approaches together and propose a quantitative extension of rewriting logic as a flexible formalism for quantitative deduction and computation.
Cite as
Besik Dundua, Georg Ehling, Santiago Escobar, Maribel Fernández, and Temur Kutsia. Quantitative Equational Rewriting. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 76:1-76:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dundua_et_al:LIPIcs.MFCS.2026.76,
author = {Dundua, Besik and Ehling, Georg and Escobar, Santiago and Fern\'{a}ndez, Maribel and Kutsia, Temur},
title = {{Quantitative Equational Rewriting}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {76:1--76:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.76},
URN = {urn:nbn:de:0030-drops-274580},
doi = {10.4230/LIPIcs.MFCS.2026.76},
annote = {Keywords: Quantitative rewriting, quantitative equational reasoning}
}
Document
Randomized and Quantum Approximate Matrix Multiplication
Abstract
The complexity of matrix multiplication is a central topic in computer science. While the focus has traditionally been on exact algorithms, a long line of literature also considers randomized algorithms, which return an approximate solution in faster time. In this work, we adopt a unifying perspective that frames these randomized algorithms in terms of mean estimation. Using it, we first give refined analyses of classical algorithms based on random walks by Cohen-Lewis (`99), and based on sketching by Sarlós (`06) and Drineas-Kannan-Mahoney (`06). We then propose an improvement on Cohen-Lewis that yields a single classical algorithm that is faster than all the other approaches, if we assume no use of (exact) fast matrix multiplication as a subroutine. Second, we demonstrate a quantum speedup on top of these algorithms by using the recent quantum multivariate mean estimation algorithm by Cornelissen-Hamoudi-Jerbi (`22).
Cite as
Simon Apers, Arjan Cornelissen, and Samson Wang. Randomized and Quantum Approximate Matrix Multiplication. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 77:1-77:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{apers_et_al:LIPIcs.MFCS.2026.77,
author = {Apers, Simon and Cornelissen, Arjan and Wang, Samson},
title = {{Randomized and Quantum Approximate Matrix Multiplication}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {77:1--77:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.77},
URN = {urn:nbn:de:0030-drops-274591},
doi = {10.4230/LIPIcs.MFCS.2026.77},
annote = {Keywords: randomized algorithms, quantum algorithms, streaming, approximate matrix multiplication, mean estimation}
}
Document
Randomized and Quantum Lifting for One-Way Conservative NOF Model
Abstract
We consider lifting theorems that transfer lower bounds for two-party communication problems to multiparty communication problems. In particular, following the deterministic Number-on-Forehead (NOF) lifting framework of Yang and Zhang, we study randomized and quantum one-way NOF lifting for composed problems F(z,𝐱) = f(z,G(𝐱)).
We work in a one-way NOF model in which only the last player’s view is restricted. The other players have their usual NOF views and communicate as usual, but the last player sees only the gadget output G(𝐱), not the gadget input 𝐱. This kind of restricted-view has appeared in the NOF literature under the name conservative model. Our main contribution is a pair of lifting theorems for this model. In the randomized setting, we show that lifting follows when each preimage G^{-1}(v), the set of gadget inputs with output v, looks pseudorandom to large cylinder intersections. In the quantum setting, we prove an analogous theorem. Equivalently, conservative protocols for F(z,𝐱) = f(z,G(𝐱)) can be converted into two-party one-way protocols for f(z,v) with comparable cost, and the error loss controlled by the corresponding pseudorandomness parameters. Thus, this restriction isolates a setting in which both randomized and quantum one-way NOF lifting can be proved by a direct simulation argument.
We prove the required pseudorandomness properties for the generalized inner product gadget over finite fields, using the multiparty character-sum bounds of Yang and Zhang, and for random gadgets, which give non-explicit lifting. As applications, Boolean Hidden Matching yields a randomized-versus-quantum separation in the conservative NOF model, and lifting INDEX gives randomized and quantum conservative NOF lower bounds for O(log n) players.
Cite as
Haoyu Wang and Pei Wu. Randomized and Quantum Lifting for One-Way Conservative NOF Model. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 78:1-78:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{wang_et_al:LIPIcs.MFCS.2026.78,
author = {Wang, Haoyu and Wu, Pei},
title = {{Randomized and Quantum Lifting for One-Way Conservative NOF Model}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {78:1--78:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.78},
URN = {urn:nbn:de:0030-drops-274604},
doi = {10.4230/LIPIcs.MFCS.2026.78},
annote = {Keywords: communication complexity, lifting theorem, number-on-forehead model}
}
Document
Regular Grammars as Effective Representations of Recognizable Sets of Series-Parallel Graphs
Abstract
Series-parallel (SP) graphs are binary edge-labeled graphs with a designated source and target vertex, built using serial and parallel composition. A set of graphs is recognizable if membership depends only on its image under a homomorphism into a finite algebra. For SP-graphs, and more generally, for graphs of bounded tree-width, recognizability coincides with definability in Counting Monadic Second-Order (CMSO) logic. Despite this strong logical characterization, the conciseness and algorithmic effectiveness of syntactic representations of recognizable sets of SP (and bounded-tree-width) graphs remain poorly understood.
Building on previously introduced regular grammars for SP-graphs, we show that recognizable sets admit concise and effective syntactic representations. The main contribution is an improved construction of finite recognizer algebras whose size is singly-exponential in the size of a regular grammar, improving upon the previously known double-exponential bound. As a consequence, the problems of intersection and language inclusion for sets represented by regular grammars are shown to be EXPTIME-complete, thus improving on a previously known 2EXPTIME upper bound.
Cite as
Marius Bozga, Radu Iosif, and Florian Zuleger. Regular Grammars as Effective Representations of Recognizable Sets of Series-Parallel Graphs. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 79:1-79:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bozga_et_al:LIPIcs.MFCS.2026.79,
author = {Bozga, Marius and Iosif, Radu and Zuleger, Florian},
title = {{Regular Grammars as Effective Representations of Recognizable Sets of Series-Parallel Graphs}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {79:1--79:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.79},
URN = {urn:nbn:de:0030-drops-274614},
doi = {10.4230/LIPIcs.MFCS.2026.79},
annote = {Keywords: Series-parallel graphs, Regular grammars, Recognizability, Decision problems}
}
Document
Satisfiability of Multivalued Circuits with Lists
Abstract
We study the problem ListCSat of satisfiability of multivalued circuits, whose input values are restricted by lists. We obtain a clear description of quasipolynomial time cases of the problem, where the time complexity is determined by the set of gates which are allowed to build such circuits. That is, algebraic structures induced by gates which allow for a quasipolynomial time algorithm decompose into a direct product L × N, where the polynomial clone of L is isomorphic to the clone of two element lattice and polynomial clone of N is isomorphic to the clone of some finite nilpotent Malcev algebra. If such a decomposition does not exist the ListCSat problem is NP-complete. The result applies to a general setting of finite algebras from congruence modular varieties. In our tractability proofs we assume certain lower bounds for the Boolean CC-circuits. We discuss the cases in which such an assumption can be avoided and polynomial/quasipolynomial time algorithms exist unconditionally.
Cite as
Paweł M. Idziak, Piotr Kawałek, Jacek Krzaczkowski, and Armin Weiß. Satisfiability of Multivalued Circuits with Lists. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 80:1-80:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{idziak_et_al:LIPIcs.MFCS.2026.80,
author = {Idziak, Pawe{\l} M. and Kawa{\l}ek, Piotr and Krzaczkowski, Jacek and Wei{\ss}, Armin},
title = {{Satisfiability of Multivalued Circuits with Lists}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {80:1--80:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.80},
URN = {urn:nbn:de:0030-drops-274624},
doi = {10.4230/LIPIcs.MFCS.2026.80},
annote = {Keywords: satisifiability, circuit satisfiability, solving equations, lists}
}
Document
Segment Watchman Routes
Abstract
Motivated by applications for robust guarding, we consider a variant of the multiple-watchmen problem that ensures that every point within a polygon P is seen from more than one direction: we search for two routes W₁,W₂, such that every point p ∈ P is contained in a segment w₁w₂ ⊆ P such that w₁ ∈ W₁ and w₂ ∈ W₂. We call such routes segment watchman routes.
We show that finding the two routes that are optimal with respect to the min-max criterion is weakly NP-hard even in simple polygons, and that finding the routes that are optimal with respect to the min-sum criterion is NP-hard in polygons with holes. Moreover, we present sufficient conditions for routes to be segment watchman routes, and provide a polynomial-time 2-approximation under both the min-max criterion and the min-sum criterion, both in simple polygons. Finally, we show how to generalize our results for k watchmen.
Cite as
Anna Brötzner, Omrit Filtser, Bengt J. Nilsson, Christian Rieck, and Christiane Schmidt. Segment Watchman Routes. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 81:1-81:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{brotzner_et_al:LIPIcs.MFCS.2026.81,
author = {Br\"{o}tzner, Anna and Filtser, Omrit and Nilsson, Bengt J. and Rieck, Christian and Schmidt, Christiane},
title = {{Segment Watchman Routes}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {81:1--81:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.81},
URN = {urn:nbn:de:0030-drops-274634},
doi = {10.4230/LIPIcs.MFCS.2026.81},
annote = {Keywords: Watchman routes, segment guarding, k-hull guarding, NP-hardness, approximation}
}
Document
Separating Feasibility and Movement in Solution Discovery: The Case of Path Discovery
Abstract
We study solution discovery, where the goal is to obtain a feasible solution to a problem from an initial configuration by a bounded sequence of local moves. In many applications, however, the graph that defines which vertex sets are feasible is not the same as the graph that governs how tokens, agents, or resources may move. Existing models such as token sliding and token jumping typically do not distinguish the problem graph and the movement graph. Motivated by this mismatch, we introduce a directed weighted two-graph model that cleanly separates feasibility from movement. A problem graph specifies the desired combinatorial objects, while a movement graph specifies admissible relocations and their costs. This yields a flexible framework that captures asymmetry, heterogeneous movement constraints, and weighted transitions, while subsuming classical discovery models as special cases.
We investigate this model through Path Discovery and Shortest Path Discovery, where the task is to realize a vertex set containing an s-t-path or a shortest s-t-path in the problem graph. These problems are particularly natural in applications, since directed and weighted shortest paths are among the most fundamental algorithmic primitives. At the same time, previous work has already shown that discovery can be computationally hard even when the underlying optimization problem is easy. Our results show that this phenomenon persists, and becomes especially rich, in the two-graph setting. We obtain a detailed complexity picture, identifying tractable cases as well as strong hardness results.
Cite as
Hanno von Bergen, Larissa Fastenau, Enna Gerhard, Nicola Lorenz, Stephanie Maaz, Amer E. Mouawad, Roman Rabinovich, Nicole Schirrmacher, Daniel Schmand, Sebastian Siebertz, and Mai Trinh. Separating Feasibility and Movement in Solution Discovery: The Case of Path Discovery. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 82:1-82:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{vonbergen_et_al:LIPIcs.MFCS.2026.82,
author = {von Bergen, Hanno and Fastenau, Larissa and Gerhard, Enna and Lorenz, Nicola and Maaz, Stephanie and Mouawad, Amer E. and Rabinovich, Roman and Schirrmacher, Nicole and Schmand, Daniel and Siebertz, Sebastian and Trinh, Mai},
title = {{Separating Feasibility and Movement in Solution Discovery: The Case of Path Discovery}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {82:1--82:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.82},
URN = {urn:nbn:de:0030-drops-274646},
doi = {10.4230/LIPIcs.MFCS.2026.82},
annote = {Keywords: solution discovery, shortest path discovery, token sliding, parameterized complexity}
}
Document
Setwise Distinguishable Permutations
Abstract
A family of permutations of [n] is called setwise distinguishable if for every permutation in the family there exists a subset of [n] whose image under this permutation differs from its image under any other permutation in the family. We prove that there exists a setwise distinguishable family of 2^{(2-o(1))⋅n} permutations of [n]. The result is optimal up to the o(1) term in the exponent and is achieved through an explicit construction. As an application, we obtain nearly tight conditional lower bounds on the kernelization complexity of graph coloring problems parameterized by the vertex-deletion distance to split graphs. This improves a result of Jansen and Kratsch (Inf. Comput., 2013).
Cite as
Ishay Haviv. Setwise Distinguishable Permutations. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 83:1-83:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{haviv:LIPIcs.MFCS.2026.83,
author = {Haviv, Ishay},
title = {{Setwise Distinguishable Permutations}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {83:1--83:9},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.83},
URN = {urn:nbn:de:0030-drops-274654},
doi = {10.4230/LIPIcs.MFCS.2026.83},
annote = {Keywords: permutations, parameterized complexity, kernelization, coloring problems}
}
Document
Sharp Thresholds for Temporal Motifs and Doubling Time in Random Temporal Graphs
Abstract
In this paper we study two natural models of random temporal graphs. In the first, the continuous model, each edge e is assigned l_e labels, each drawn uniformly at random from (0,1], where the numbers l_e are independent random variables following the same discrete probability distribution. In the second, the discrete model, the l_e labels of each edge e are chosen uniformly at random from a set {1,2,…,T}. In both models we study the existence of δ-temporal motifs. Here a δ-temporal motif consists of a pair (H,P), where H is a fixed static graph and P is a partial order over its edges. A temporal graph 𝒢 = (G,λ) contains (H,P) as a δ-temporal motif if 𝒢 has a simple temporal subgraph on the edges of H whose time labels are ordered according to P, and whose life duration is at most δ. We prove sharp existence thresholds for all δ-temporal motifs, and we identify a qualitatively different behavior from the analogous static thresholds in Erdős-Rényi random graphs. Applying the same techniques, we then characterize the growth of the largest δ-temporal clique in the continuous variant of our random temporal graphs model. Finally, we consider the doubling time of the reachability ball centered on a small set of vertices of the random temporal graph as a natural proxy for temporal expansion. We prove sharp upper and lower bounds for the maximum doubling time in the continuous model.
Cite as
Henry Austin, George B. Mertzios, and Paul G. Spirakis. Sharp Thresholds for Temporal Motifs and Doubling Time in Random Temporal Graphs. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 84:1-84:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{austin_et_al:LIPIcs.MFCS.2026.84,
author = {Austin, Henry and Mertzios, George B. and Spirakis, Paul G.},
title = {{Sharp Thresholds for Temporal Motifs and Doubling Time in Random Temporal Graphs}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {84:1--84:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.84},
URN = {urn:nbn:de:0030-drops-274660},
doi = {10.4230/LIPIcs.MFCS.2026.84},
annote = {Keywords: Random temporal graph, \delta-temporal motif, sharp upper and lower bounds, doubling time}
}
Document
Simple Nash Equilibria for Qualitative Multiplayer Games
Abstract
We investigate memory requirements for Nash and subgame-perfect equilibria in turn-based deterministic games with ω-regular objectives. We prove that memoryless randomised (i.e., stationary) subgame-perfect equilibria always exist in games with reachability, safety, and 0-2 Muller objectives (i.e., Muller objectives for which accepting sets are either up- or downward closed), and any combination of these objectives. We provide an algorithm to construct such an equilibrium. We also show that randomisation may be required to construct memoryless equilibria in games with reachability or Büchi as well as safety or CoBüchi objectives, and that memoryless equilibria need not exist for any other class of Muller objectives (with respect to the Mostowski hierarchy).
Cite as
Mona Alluwaym, James C. A. Main, and Sven Schewe. Simple Nash Equilibria for Qualitative Multiplayer Games. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 85:1-85:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{alluwaym_et_al:LIPIcs.MFCS.2026.85,
author = {Alluwaym, Mona and Main, James C. A. and Schewe, Sven},
title = {{Simple Nash Equilibria for Qualitative Multiplayer Games}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {85:1--85:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.85},
URN = {urn:nbn:de:0030-drops-274679},
doi = {10.4230/LIPIcs.MFCS.2026.85},
annote = {Keywords: games on graphs, multiplayer games, Nash equilibria, subgame-perfect equilibria, memoryless strategies}
}
Document
Smallest Suffixient Set Maintenance in Near-Real-Time
Abstract
The size of the smallest suffixient set of positions of a string recently emerged as a new measure of string repetitiveness - a measure reflecting how much of repetitive content the string contains. We study how to maintain the smallest suffixient set online in near-real-time, that is with small (in our case, polyloglog) worst-case time for processing each letter. Two frameworks are considered: when the text is given letter-by-letter in either right-to-left or left-to-right order. Our central algorithmic tool is Weiner’s suffix tree algorithm and associated algorithmic primitives for its efficient implementation.
Cite as
Dominik Köppl and Gregory Kucherov. Smallest Suffixient Set Maintenance in Near-Real-Time. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 86:1-86:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{koppl_et_al:LIPIcs.MFCS.2026.86,
author = {K\"{o}ppl, Dominik and Kucherov, Gregory},
title = {{Smallest Suffixient Set Maintenance in Near-Real-Time}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {86:1--86:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.86},
URN = {urn:nbn:de:0030-drops-274684},
doi = {10.4230/LIPIcs.MFCS.2026.86},
annote = {Keywords: online algorithms, string algorithms, suffix tree, real-time computation, smallest suffixient set, string attractor}
}
Document
Space Complexity of Reachability in Simple Path Graphs
Abstract
One of the central questions in space complexity is whether nondeterministic logspace computations (NL) can be simulated in unambiguous logspace (UL). A stronger notion, reach unambiguity (ReachUL ⊆ UL ∩ coUL), requires every configuration reachable from the start to have exactly one computation path [Buntrock et al., 1991]. It is known that directed graph reachability is NL-complete, planar graph reachability is in UL, and undirected graph reachability is in deterministic logspace (L). In this work, we study the space complexity of reachability problem for restricted directed graph families and show the following.
1) For reach unambiguous graphs (graphs with at most one path from start vertex to every vertex), [Lange, 1997] showed that reachability is in ReachUL. As our main result, we show that for simple path graphs (introduced in [Kannan et al., 2008], which contains reach unambiguous graphs), where each vertex has at most one simple path from the start, the reachability problem lies in UL ∩ coUL. The key difficulty lies in recognizing whether the input graph is a simple path graph or not.
2) Our first result can also be equivalently stated as follows: the recognition problem for simple path graphs is in UL ∩ coUL if and only if the reachability problem restricted to simple path graphs is also in UL ∩ coUL. Inspired by this, we investigate the complexity of graph recognition versus graph reachability for other directed graph classes. Observe that for any graph class, solving reachability (for the class) also solves the recognition problem for that class. We show that for reach unambiguous graphs, solving recognition is as hard as solving reachability (making both of them ReachUL-complete).
Cite as
Krishnamoorthy Dinesh and Chandana Sasidharan. Space Complexity of Reachability in Simple Path Graphs. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 87:1-87:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dinesh_et_al:LIPIcs.MFCS.2026.87,
author = {Dinesh, Krishnamoorthy and Sasidharan, Chandana},
title = {{Space Complexity of Reachability in Simple Path Graphs}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {87:1--87:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.87},
URN = {urn:nbn:de:0030-drops-274695},
doi = {10.4230/LIPIcs.MFCS.2026.87},
annote = {Keywords: Space complexity, Graph reachability, Simple path graphs, Reach unambiguity, Unambiguity, UL}
}
Document
Structural Parameterizations of Geodetic Set on Directed (Acyclic) Graphs
Abstract
In Directed Geodetic Set, we are given a (directed) graph and seek a small solution set S ⊆ V(G) such that every vertex lies on a shortest directed path between two vertices in S. While most prior work on Directed Geodetic Set has focused on undirected graphs, in this article we study the problem on directed graphs from the perspective of parameterized complexity.
It is known that the problem is W[2]-hard when parameterized by the solution size k, even on directed acyclic graphs (DAGs). We investigate structural parameterizations of the problem.
Our first result is a kernel of size 2^O(vcn) for Directed Geodetic Set on general digraphs, where vcn denotes the vertex cover number of the underlying (undirected) graph. This implies an algorithm running in time 2^O(vcn²) ⋅ n^O(1). Furthermore, we prove that, assuming the ETH, the problem does not admit an algorithm running in time 2^o(vcn²) ⋅ n^O(1). Such a tight quadratic exponential lower bound in the parameter is relatively uncommon in parameterized complexity. These results generalize earlier work on undirected graphs by Foucaud et al. [STACS 2025], and complements a recent result on directed graph by Foucaud et al. [CALDAM 2026], that showed that the problem is para-NP-hard for the pathwidth and feedback vertex set number of the underlying graph.
Next, we show that on general digraphs, Directed Geodetic Set admits a natural kernel of size (kΔ)^O(rdiam), where Δ is the maximum degree and rdiam denotes the reachability diameter of the digraph (a natural analogue of diameter of undirected graphs). This yields an algorithm running in time (kΔ)^O(rdiam⋅k) ⋅ n^O(1). We further prove that, assuming the ETH, the problem does not admit an algorithm running in time (kΔ)^o(rdiam ⋅ k) ⋅ n^O(1). Finally, we justify the necessity of combining parameters by establishing the following hardness results for Directed Geodetic Set:
1) It is W[2]-hard parameterized by k, even on digraphs of maximum degree 3.
2) It is para-NP-hard parameterized by maximum degree and reachability diameter. One can infer that the problem remains W[2]-hard when parameterized by k, even on graphs of reachability diameter 3 from Araújo and Arraes [DAM 2022].
All our conditional lower bounds and hardness results hold even when the input digraph is restricted to be a DAG.
Cite as
Laurent Beaudou, Florent Foucaud, Lucas Lorieau, and Prafullkumar Tale. Structural Parameterizations of Geodetic Set on Directed (Acyclic) Graphs. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 88:1-88:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{beaudou_et_al:LIPIcs.MFCS.2026.88,
author = {Beaudou, Laurent and Foucaud, Florent and Lorieau, Lucas and Tale, Prafullkumar},
title = {{Structural Parameterizations of Geodetic Set on Directed (Acyclic) Graphs}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {88:1--88:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.88},
URN = {urn:nbn:de:0030-drops-274708},
doi = {10.4230/LIPIcs.MFCS.2026.88},
annote = {Keywords: Geodetic Set, Directed Graphs, NP-hardness, Parameterized Complexity}
}
Document
Sublinear Time Algorithms for Abelian Group Property Testing
Abstract
In this paper, we study the problems of abelian group property testing in two models. In the partially specified model (PS-model), the algorithm does not know the group size but can access randomly chosen elements of the group, along with the Cayley table of these elements, which provides the result of the binary operation for every pair of selected elements. In the stronger fully specified model (FS-model), the algorithm knows the size of the group and has access to all its elements and the Cayley table.
In property testing of abelian group property, given a finite set G and oracle access to a binary operation *:G² → G, we aim to distinguish whether (G,*) is an abelian group or is ε-far from any abelian group over G.
Using a novel approach, we present a tester in the PS-model (and consequently in the FS-model) that runs in time Õ(√{|G|} + 1/ε), improving upon the Goldreich-Tauber tester, which runs in time O(|G|/ε). Additionally, our tester improves another tester by Goldreich and Tauber that runs in time O(|G|²) and makes Õ(|G| + 1/ε) queries.
We further extend our result to testing subclasses of abelian groups G that are closed under isomorphism. Specifically, if one can decide in time T whether an abelian group of the form ℤ_{m_1} × ⋯ × ℤ_{m_r} belongs to G, then there exists a tester for G that runs in time T+Õ(√{|G|} + 1/ε) and makes O(√{|G|} + 1/ε) queries. This result gives testers that run in time O(√{|G|} + 1/ε) for subclasses such as abelian groups of rank at most k, abelian p-groups, and vector spaces over ℤ_p.
We then present two subclasses, G₁ and G₂, of abelian groups that are closed under isomorphism for which any tester for G₁ in the FS-model must run in time Ω(|G|^{1/4} + 1/ε), and any tester for G₂ in the PS-model must run in time Ω(√{|G|} + 1/ε), showing that our approach provides tight bounds for certain subclasses of abelian groups.
Cite as
Nader H. Bshouty. Sublinear Time Algorithms for Abelian Group Property Testing. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 89:1-89:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bshouty:LIPIcs.MFCS.2026.89,
author = {Bshouty, Nader H.},
title = {{Sublinear Time Algorithms for Abelian Group Property Testing}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {89:1--89:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.89},
URN = {urn:nbn:de:0030-drops-274716},
doi = {10.4230/LIPIcs.MFCS.2026.89},
annote = {Keywords: Property testing, Abelian group}
}
Document
Testing Equivalence to the Hamiltonian Cycle Polynomial
Abstract
The Hamiltonian Cycle polynomial, denoted as HC_n, is defined to be the sum of the weighted Hamiltonian Cycles in an n-vertex complete digraph, with vertices labeled 1 to n and edges weighted by formal variables x_{i,j}. The Permanent and HC, defined as the family {HC_n | n ≥ 1}, were studied by Valiant (STOC 1979), with the former shown to be VNP-complete over all fields of characteristic other than 2, and the latter to be VNP-complete over every field. Since its introduction, HC has been studied from the perspective of circuit lower bounds by Jerrum-Snir (JACM 1982), determinantal complexity by Huttenhain-Ikenmeyer (LAA 2016), and its connection with the Permanent and the Determinant polynomials by Goulden-Jackson (EJC 1981) and Grochow (ToC 2017). It has been the most prominent choice for generalising results to all fields, such as in Malod (CCC 2007) and Grochow-Mulmuley-Qiao (ICALP 2016), owing to its VNP-completeness over every field. Hrubes (ToCT, 2016) showed the VNP-completeness of many graph-based polynomial families over every field by using HC.
In Kayal (STOC 2012), a randomised polynomial time algorithm was given for the following problem: Given an n²-variate degree-n polynomial f(𝐱) ∈ 𝔽[𝐱] as a black box, decide if there exists A ∈ GL_{n²}(𝔽) such that f(𝐱) = Perm_n(A𝐱). Here, the Permanent polynomial Perm_n computes the permanent of the n × n symbolic matrix (x_{i,j}). This problem is known as testing equivalence to the Permanent, or alternatively, ET for Permanent.
In this work, we study ET for HC. While both families are VNP-complete, the efficient ET algorithm for Permanent does not imply the same for HC. Besides, there are crucial differences between the two polynomials that make studying the complexity of ET for HC interesting: The underlying decision problem corresponding to the Permanent is in P (detecting perfect matchings in a bipartite graph), but that for HC (detecting Hamiltonian cycles in a digraph) is NP-complete. The Permanent polynomial is known to be characterised by its symmetries as shown by Mulmuley-Sohoni (SIAM J. Computing, 2001). This property yields an efficient algorithm for the circuit-testing problem for the Permanent, a special case of ET for the Permanent, in which we check whether a given circuit computes the Permanent. In contrast, we show HC_n is not characterised by its symmetries.
In this work, we give a randomised polynomial time ET algorithm for HC with mild constraints on the underlying field. The algorithm is obtained by studying and completely characterising the Lie algebra and the symmetries of HC_n. We show that, like the Permanent polynomial, the symmetries of HC_n are generated by permutation and scaling matrices over large enough fields. However, we also show that, unlike the Permanent polynomial, HC_n is not characterised by its symmetries. Nevertheless, like the Permanent polynomial, HC_n is downward self-reducible, as shown in Zhang-Bai (TCS 2011), which implies HC_n is characterised by circuit identities and that we can efficiently test whether a given circuit C computes HC_n. We also get a Flip theorem for HC_n as a result of its circuit identities.
Cite as
Agrim Dewan. Testing Equivalence to the Hamiltonian Cycle Polynomial. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 90:1-90:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dewan:LIPIcs.MFCS.2026.90,
author = {Dewan, Agrim},
title = {{Testing Equivalence to the Hamiltonian Cycle Polynomial}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {90:1--90:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.90},
URN = {urn:nbn:de:0030-drops-274729},
doi = {10.4230/LIPIcs.MFCS.2026.90},
annote = {Keywords: Equivalence Testing, Hamiltonian Cycle Polynomial, Symmetries, Lie Algebra, Circuit identities}
}
Document
The 2D Ray Tracing Problem Using ABCD Lenses and Mirrors Is Turing Complete
Abstract
We establish that the two-dimensional ray tracing problem with thin lenses and plane mirrors is Turing-complete, thereby resolving an open question posed by Reif et al. in 1994 as to whether three-dimensional space is necessary for computational universality in optical systems. To this end, we consider the standard approximation of reflection and refraction, namely the ABCD model for paraxial optics, which describes ray propagation through lenses (refraction) via a 2 × 2 matrix, combined with the geometric reflection model for plane mirrors.
In the absence of mirrors, two-dimensional ray tracing using any combination of lenses in this ABCD matrix model can be described by a single 2 × 2 matrix–vector product, where the matrix has real entries and determinant 1. Conversely, we show that any such matrix with determinant 1 can be represented as a composition of exactly three appropriately spaced thin lenses. When mirrors are combined with lenses, the ray tracing problem can be described by a flowchart using only two variables, which establishes Turing computability for rational-valued inputs, spaces and matrix entries. Building on this observation, we present a construction of ray tracing that simulates a reversible Turing machine.
We begin with a restricted version of the reversible flowchart problem, in which only two variables and certain linear functions are permitted. We prove that this restricted variant is Turing-complete. We then show that such a flowchart admits a geometric realization using lenses and mirrors in our model, thereby establishing the main result: Turing-completeness of the two-dimensional ray tracing problem with ABCD-model lenses and mirrors.
Cite as
Rosemary U. Adejoh, Andreas Jakoby, Sneha Mohanty, and Christian Schindelhauer. The 2D Ray Tracing Problem Using ABCD Lenses and Mirrors Is Turing Complete. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 91:1-91:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{adejoh_et_al:LIPIcs.MFCS.2026.91,
author = {Adejoh, Rosemary U. and Jakoby, Andreas and Mohanty, Sneha and Schindelhauer, Christian},
title = {{The 2D Ray Tracing Problem Using ABCD Lenses and Mirrors Is Turing Complete}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {91:1--91:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.91},
URN = {urn:nbn:de:0030-drops-274735},
doi = {10.4230/LIPIcs.MFCS.2026.91},
annote = {Keywords: Turing completeness, optical computation, ray tracing, ABCD matrix, thin lenses, plane mirrors, reversible Turing machine, flowchart, reversible flowchart}
}
Document
The Complexity of Edge-Induced Greedy Subgraph Building Algorithms Within P
Abstract
A common approach used to efficiently solve problems is to develop sequential greedy algorithms. Such algorithms are easily implemented and provide polynomial-time solutions. A natural next step towards building more efficient algorithms is to develop parallel algorithms. However, sequential greedy algorithms seldom lead to parallel algorithms; computing the output of sequential greedy algorithms is often shown to be P-complete and thus "inherently sequential" under the commonly believed assumption that P ≠ NC, where NC is the class of efficiently parallelizable problems.
Greedy edge-induced (resp., vertex-induced) subgraph building algorithms for a property π operate like so. For given graph G, a subgraph of G is built by adding edges (resp., vertices) in a given order unless the inclusion of said edge (resp., vertex) would contradict property π within the subgraph. For vertex-induced greedy subgraph building algorithms, Miyano (1989) provided a comprehensive result: computing the subgraph output by such algorithms is typically P-complete. In contrast, little is known about its edge-induced counterpart. In this work, we analyze the complexity of the Lexicographically First Maximal H-free edge-induced subgraph problem, which is concerned with computing the output of greedy edge-induced subgraph building algorithms where the property π is that the subgraph is H-free. This gives us insight into the largely overlooked edge-induced versions of greedy subgraph building algorithms and into how graph structure influences the complexity of such algorithms.
Our primary contribution is a trichotomy theorem for the cases where H is a tree: we show that the problem is either P-complete, CC-complete, or in L, where CC is the class of problems solvable using comparator circuits - or, equivalently, problems reducible to the lexicographically first maximal matching problem. In contrast, the vertex-induced version is either P-complete or in L, and such dichotomy theorems are much more common. Our additional technical contributions include: (1) an iterative approach to hardness proofs by focusing on a set of "smaller" problems and extending hardness via simple constructions, and (2) expanding on the scarce set of problems known to be CC-complete.
Cite as
Zohair Raza Hassan and Edith Hemaspaandra. The Complexity of Edge-Induced Greedy Subgraph Building Algorithms Within P. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 92:1-92:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hassan_et_al:LIPIcs.MFCS.2026.92,
author = {Hassan, Zohair Raza and Hemaspaandra, Edith},
title = {{The Complexity of Edge-Induced Greedy Subgraph Building Algorithms Within P}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {92:1--92:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.92},
URN = {urn:nbn:de:0030-drops-274745},
doi = {10.4230/LIPIcs.MFCS.2026.92},
annote = {Keywords: P-completeness, parallelizability, lexicographically first edge problems}
}
Document
The Descriptive Complexity of Relation Modification Problems
Abstract
A relation modification problem gets a logical structure and a natural number k as input and asks whether k modifications of the structure suffice to make it satisfy a predefined property. We provide a complete classification of the classical and parameterized complexity of relation modification problems - the latter w. r. t. the modification budget k - based on the descriptive complexity of the respective target property. We consider different types of logical structures on which modifications are performed: Whereas monadic structures and undirected graphs without self-loops each yield their own complexity landscapes, we find that modifying undirected graphs with self-loops, directed graphs, or arbitrary logical structures is equally hard w. r. t. quantifier patterns.
Moreover, we observe that all classes of problems considered in this paper are subject to a strong dichotomy in the sense that they are either very easy to solve (that is, they lie in para-AC^{0↑} or TC^0) or intractable (that is, they contain W[2]-hard or NP-hard problems).
Cite as
Florian Chudigiewitsch, Marlene Gründel, Christian Komusiewicz, Nils Morawietz, and Till Tantau. The Descriptive Complexity of Relation Modification Problems. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 93:1-93:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chudigiewitsch_et_al:LIPIcs.MFCS.2026.93,
author = {Chudigiewitsch, Florian and Gr\"{u}ndel, Marlene and Komusiewicz, Christian and Morawietz, Nils and Tantau, Till},
title = {{The Descriptive Complexity of Relation Modification Problems}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {93:1--93:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.93},
URN = {urn:nbn:de:0030-drops-274752},
doi = {10.4230/LIPIcs.MFCS.2026.93},
annote = {Keywords: graph problems, descriptive complexity, edge modification, parameterized complexity, circuit complexity}
}
Document
The Entailment Problem for Separation Logic with Overlaid Structures
Abstract
Separation Logic (SL) enables reasoning about programs that manipulate pointers. Its key feature is the separating conjunction ⋆, which asserts that two formulas hold on disjoint portions of memory. We consider an extension of SL, called Overlaid SL (OSL), that allows non-disjoint combinations of data structures defined over different fields, enriched with set constraints on the nodes of these structures. We prove that entailment is decidable for a broad class of data structures satisfying the so-called PCE conditions of [Iosif et al., 2013], thus extending this result to OSL. Our decision procedure is nondeterministic with doubly exponential time complexity.
Cite as
Lucas Bueri, Nicolas Peltier, Quentin Petitjean, and Mihaela Sighireanu. The Entailment Problem for Separation Logic with Overlaid Structures. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 94:1-94:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bueri_et_al:LIPIcs.MFCS.2026.94,
author = {Bueri, Lucas and Peltier, Nicolas and Petitjean, Quentin and Sighireanu, Mihaela},
title = {{The Entailment Problem for Separation Logic with Overlaid Structures}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {94:1--94:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.94},
URN = {urn:nbn:de:0030-drops-274766},
doi = {10.4230/LIPIcs.MFCS.2026.94},
annote = {Keywords: Decision Procedure, Separation Logic, Inductive Definitions, Overlaid Data Structures}
}
Document
The Parameterized Complexity of Maximum Span on Natural Matroid Classes
Abstract
We study Maximum Span, motivated by the recent Maximum Span Hypothesis of Karthik and Khot [SODA 2025], which suggests strong parameterized intractability for finding large structured subsets in vector spaces. Formally, given a matrix M and integers k and t, the task is to decide whether there exists a linearly independent set S of at most k columns such that at least t additional columns of M lie in span(S). Equivalently, the goal is to identify a low-rank witness whose span covers many input columns.
We initiate a systematic study of the parameterized complexity of Maximum Span on natural matroid classes, revealing a diverse complexity landscape. We first show that the problem is polynomial-time solvable on laminar matroids, via a dynamic program over the laminar tree. In sharp contrast, on graphic matroids the problem is W[1]-hard parameterized by k+t, and, assuming Gap-ETH, admits no f(k)⋅ n^𝒪(1)-time k^o(1)-approximation. On cographic matroids, we show that the problem is equivalent to deleting at most k+t edges so as to create at least t+1 connected components; this yields fixed-parameter tractability parameterized by k+t, and W[1]-hardness parameterized by t. On transversal matroids, using a Hall-type interpretation, we prove W[1]-hardness parameterized by k+t.
For strict gammoids, we develop a separator-based formulation. We prove W[1]-hardness parameterized by k+t, give an XP algorithm parameterized by t, and obtain FPT 2^k-approximation algorithms in both the directed and undirected settings. For general gammoids, we establish W[1]-hardness parameterized by k+t, NP-hardness already for t = 1, and an XP algorithm parameterized by k. Together, these results give a detailed parameterized complexity map for Maximum Span across fundamental matroid classes, ranging from polynomial-time solvability to fixed-parameter algorithms, XP algorithms, approximation algorithms, and strong hardness.
Cite as
Madhumita Kundu, Ashutosh Rai, Sahiba, and Saket Saurabh. The Parameterized Complexity of Maximum Span on Natural Matroid Classes. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 95:1-95:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kundu_et_al:LIPIcs.MFCS.2026.95,
author = {Kundu, Madhumita and Rai, Ashutosh and Sahiba and Saurabh, Saket},
title = {{The Parameterized Complexity of Maximum Span on Natural Matroid Classes}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {95:1--95:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.95},
URN = {urn:nbn:de:0030-drops-274775},
doi = {10.4230/LIPIcs.MFCS.2026.95},
annote = {Keywords: Fixed Parameter Tractability, W\lbrack1\rbrack-hardness, FPT Approximation, Graphic Matroids, Cographic Matroids, Transversal Matroids, Strict Gammoids, Gammoids, Laminar Matroids}
}
Document
The Polynomial Hierarchy and ω-Categorical CSPs
Abstract
In 2008, Bodirsky and Grohe showed that for every Π_n^P-level of the Polynomial Hierarchy (PH) there are ω-categorical Constraint Satisfaction Problems (CSPs) complete for this level. We show that, in fact, there are ω-categorical CSPs complete for any level of the PH. To this end, we use a recent result of Bodirsky, Knäuer, and Rudolph for constructing ω-categorical CSPs from sentences of Monadic Second-Order logic (MSO) with certain preservation properties. As a secondary contribution, we develop a new tool for producing MSO sentences satisfying said preservation properties.
Cite as
Santiago Guzmán-Pro and Jakub Rydval. The Polynomial Hierarchy and ω-Categorical CSPs. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 96:1-96:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{guzmanpro_et_al:LIPIcs.MFCS.2026.96,
author = {Guzm\'{a}n-Pro, Santiago and Rydval, Jakub},
title = {{The Polynomial Hierarchy and \omega-Categorical CSPs}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {96:1--96:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.96},
URN = {urn:nbn:de:0030-drops-274789},
doi = {10.4230/LIPIcs.MFCS.2026.96},
annote = {Keywords: monadic second-order logic, constraint satisfaction, homomorphism-closed, \omega-categoricity, polynomial hierarchy, conjunctive query, primitive positive formula, quantifiers}
}
Document
The Power of Small Symmetries
Abstract
Resolution with symmetries is a natural extension of the Resolution proof system that allows to use symmetries of the formula to simplify the proof. Symmetries can be global (applied to the whole input formula), local (applied to a subformula), or dynamic (applied to newly derived clauses as well). The framework of Resolution with (global) symmetries was introduced by [Krishnamurthy, 1985] and further extended by [Arai and Urquhart, 2000] to local symmetries. Later, [Szeider, 2005] generalized this approach to homomorphisms and introduced the notion of Resolution with dynamic symmetries. While proving superpolynomial proof-size lower bounds for Resolution with dynamic symmetries remains an open problem already for two decades, the power of proof systems with global and local symmetries is well studied: exponential lower bounds have been proven for these proof systems, as well as exponential separations between all of them. However, these systems are too general to reflect practical applications since it is computationally too hard to find and efficiently exploit arbitrary symmetries.
In this work, we introduce the notion of small symmetries: symmetries that can operate on a limited number of variables at the same time. Resolution with small symmetries gives hopes both for practical applications and for theoretical study of dynamic symmetries. We show that proof systems with both local and global small symmetries form strict hierarchies w.r.t. the size of symmetries. We prove exponential separations between proof systems with symmetries of different sizes and types. It turns out that even lower levels of these hierarchies are exponentially separated from Resolution and stronger proof systems, such as constant-depth Frege. As a byproduct of our constructions, we obtain an exponential separation between the classical systems SRCI and SRII that was not known before.
Cite as
Nikita Gaevoy. The Power of Small Symmetries. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 97:1-97:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gaevoy:LIPIcs.MFCS.2026.97,
author = {Gaevoy, Nikita},
title = {{The Power of Small Symmetries}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {97:1--97:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.97},
URN = {urn:nbn:de:0030-drops-274799},
doi = {10.4230/LIPIcs.MFCS.2026.97},
annote = {Keywords: proof complexity, complexity lower bounds, resolution with symmetries, small symmetries}
}
Document
Towards a Universal Gateset for QMA₁
Abstract
QMA₁ is QMA with perfect completeness: in the YES-case the verifier must accept some proof with probability exactly 1. Whether QMA₁ = QMA remains open, and even the definition of QMA₁ has a gateset issue, since Solovay-Kitaev only gives approximate synthesis and may destroy perfect completeness. For a gateset 𝒢, we write QMA₁^𝒢 for QMA₁ restricted to verifiers using gates from 𝒢.
Using the gatesets 𝒢_{2^k} of Amy et al. (RC 2024), we prove that QMA₁^𝒢 ⊆ QMA₁^{𝒢_{2^k}} for every finite gateset 𝒢 whose entries lie in the cyclotomic field ℚ(ζ_{2^k}), ζ_{2^k} = e^{2πi/2^k}. For BQP₁ (aka coRQP), the rational gateset 𝒢₂ already suffices for all these fields. We also give complete problems for the resulting classes: quantum 𝓁-SAT over ℚ(ζ_{2^k}) is complete for QMA₁^{𝒢_{2^k}} for all 𝓁 ≥ 4, and also for 𝓁 = 3 when k ≥ 3.
The main technical tool is to use linear combinations of unitaries and postselection to exactly apply operators whose entries lie in the relevant field, and then use oblivious amplitude amplification to make the postselection failure probability negligible. This also gives an exact kernel test for Hamiltonians. As a consequence, we prove the first QMA₁-complete 2-local Hamiltonian problem: for k ≥ 3, deciding whether a 2-local Hamiltonian H over ℚ(ζ_{2^k}) has σ₁(H) = 0 or σ₁(H) ≥ 1/poly is complete for QMA₁^{𝒢_{2^k}}. The same ideas extend to sparse Hamiltonians and yield the first QMA₁(2)-complete Hamiltonian problem.
Finally, we apply the gateset framework to clique homology. We prove that the Gapped Clique Homology problem on weighted graphs defined by King and Kohler (FOCS 2024) is QMA₁^𝒢₂-complete, and the Clique Homology problem (Kaibel and Pfetsch, 2002) without promise gap is PSPACE-complete, resolving a conjecture of Crichigno and Kohler (Nat. Commun. 2024).
Cite as
Dorian Rudolph. Towards a Universal Gateset for QMA₁. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 98:1-98:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{rudolph:LIPIcs.MFCS.2026.98,
author = {Rudolph, Dorian},
title = {{Towards a Universal Gateset for QMA₁}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {98:1--98:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.98},
URN = {urn:nbn:de:0030-drops-274802},
doi = {10.4230/LIPIcs.MFCS.2026.98},
annote = {Keywords: QMA with perfect completeness, quantum satisfiability, universal gatesets, local Hamiltonian, clique homology}
}
Document
Upper Clique Transversal on Interval Graphs and Beyond
Abstract
The Upper Clique Transversal (UCT) problem asks for the size of the largest minimal set of vertices intersecting all the maximal cliques of the input graph. This problem was recently introduced by Milanič and Uno [WG 2023], who studied its complexity on several graph classes. They showed that the problem is NP-hard on chordal graphs, and gave polynomial-time algorithms for UCT on split and on proper interval graphs. They left open the complexity of UCT on interval graphs. In this work we settle this question by giving a polynomial-time algorithm for UCT on interval graphs. We show that even on the more general class of rooted directed path graphs, which can be understood as a "tree-like version" of interval graphs, the problem remains polynomial-time solvable. On the negative side, we observe as consequences of the NP-hardness proof for chordal graphs due to Milanič and Uno that the problem is NP-hard on graphs of path-independence number two (interval graphs have path-independence number one) and on well-partitioned chordal graphs which lie between split and chordal graphs.
Cite as
Lars Jaffke, Paloma de Lima, and Amir Nikabadi. Upper Clique Transversal on Interval Graphs and Beyond. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 99:1-99:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{jaffke_et_al:LIPIcs.MFCS.2026.99,
author = {Jaffke, Lars and de Lima, Paloma and Nikabadi, Amir},
title = {{Upper Clique Transversal on Interval Graphs and Beyond}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {99:1--99:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.99},
URN = {urn:nbn:de:0030-drops-274818},
doi = {10.4230/LIPIcs.MFCS.2026.99},
annote = {Keywords: interval graphs, rooted directed path graphs, clique transversal}
}
Document
When Darwin Met Ianus: Dichotomies of Expressivity
Abstract
The classifications of temporal and phylogeny constraint languages stand among the most seminal complexity classifications within infinite-domain Constraint Satisfaction Problems (CSPs), yet remain the most mysterious in terms of algorithms and algebraic invariants for the tractable cases. We show that those languages which do not pp-construct EVERYTHING (and thus by the classifications are solvable in polynomial time) have, in fact, very limited expressive power as measured by the graphs and hypergraphs they can pp-interpret. This limitation yields many previously unknown algebraic consequences, while also providing new, uniform proofs for known invariance properties. In particular, we show that such temporal and phylogeny constraint languages admit 4-ary pseudo-Siggers polymorphisms - a result that sustains the possibility that the existence of such polymorphisms extends to the much broader context of the Bodirsky-Pinsker conjecture. Although temporal and phylogeny constraint languages appear to follow fundamentally different algorithmic principles, our proofs reveal a common core and proceed along strikingly similar lines.
When Ianus can't express it all
He tries in vain, he hits a wall
When for 𝕂₃ no way he knows
His face of pseudo-loops he shows.
As Darwin finds such twisted edge
To pines and vines he makes this pledge:
"Should free of pseudo-loops you shine
All finite structures shall be thine!"
Cite as
Johanna Brunar, Michael Pinsker, and Moritz Schöbi. When Darwin Met Ianus: Dichotomies of Expressivity. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 100:1-100:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{brunar_et_al:LIPIcs.MFCS.2026.100,
author = {Brunar, Johanna and Pinsker, Michael and Sch\"{o}bi, Moritz},
title = {{When Darwin Met Ianus: Dichotomies of Expressivity}},
booktitle = {51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
pages = {100:1--100:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-442-0},
ISSN = {1868-8969},
year = {2026},
volume = {386},
editor = {Kouck\'{y}, Michal and Petrișan, Daniela},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.100},
URN = {urn:nbn:de:0030-drops-274824},
doi = {10.4230/LIPIcs.MFCS.2026.100},
annote = {Keywords: Constraint Satisfaction Problem (CSP), Temporal Constraint language, Phylogeny Constraint language, pseudo-loop, primitive positive interpretation, polymorphism, oligomorphic permutation group, identity}
}