Volume
LIPIcs, Volume 345
50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)
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Series:
Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Mathematical Foundations of Computer Science (MFCS)
Event
MFCS 2025, August 25-29, 2025, Warsaw, PolandEditors
Publication Details
- published at: 2025-08-20
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-388-1
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Complete Volume
LIPIcs, Volume 345, MFCS 2025, Complete Volume
Abstract
LIPIcs, Volume 345, MFCS 2025, Complete Volume
Cite as
50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 1-1650, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@Proceedings{gawrychowski_et_al:LIPIcs.MFCS.2025,
title = {{LIPIcs, Volume 345, MFCS 2025, Complete Volume}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {1--1650},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025},
URN = {urn:nbn:de:0030-drops-244417},
doi = {10.4230/LIPIcs.MFCS.2025},
annote = {Keywords: LIPIcs, Volume 345, MFCS 2025, Complete Volume}
}
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Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 0:i-0:xviii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{gawrychowski_et_al:LIPIcs.MFCS.2025.0,
author = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {0:i--0:xviii},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.0},
URN = {urn:nbn:de:0030-drops-244400},
doi = {10.4230/LIPIcs.MFCS.2025.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Invited Talk
Lambdas, Transducers and MSO (Invited Talk)
Abstract
This talk will revolve around the classical question:
What kind of maps from words to words or trees to trees can be considered as "well behaved" from an automata theoretic point of view?
What means to be well-behaved is left unspecified on purpose, but one thing is sure is that one would expect as minimal requirement that the inverse image of a regular language under such a map be effectively regular. One would also expect to have nice closure properties for these maps, in particular under composition. Ideally, we would also like to have several different equivalent computation models for representing them.
This talk advertises the use of simply-typed lambda calculus as a good way to approach this question, and presents newer results concerning translations from and to exponential growth maps.
This line of research is rooted in a long literature, starting from the works of Damm on higher order grammars, and the works of Engelfriet and Vogler on macro-tree transducers, and more generally from works on higher-order tree transducers, as well as the results of Ong et al. on decidability of MSO over higher-order recursion schemes (a model that produces infinite trees). These works show in a broad sense that models of computation defined in simply typed lambda-calculus do preserve regularity under taking the inverse image.
Another connection is the recent implicit automata research programme initiated by Nguy~ên and Pradic, that was inspired by the seminal work of Hillebrand and Kanellakis. Here, automata models a directly viewed as programs written in some form of typed lambda-calculus, using Church encoding for representing inputs and outputs. When appropriately controlling the type system various it is possible to characterize various classes of finite state transducers studied in the literature.
A last approach comes from finite model-theory. In this context, Monadic Second-Order Logic (MSO for short, the extension of first-order in which one can further quantify over sets) is the prime logic for describing regular languages over words or trees. This logic can also be used for defining transformations from words/trees to words/trees. These are the notions of MSO-interpretations or MSO-transducers. A major contribution here is the study of the expressive power of MSO-interpretations of polynomial growth from words to words that was undertaken by Bojańczyk, Kiefer and Lhote, and shown equivalent to other formalisms of transducers and programming languages. This is the class of polyregular functions, and it satisfies all the expectation of the question. Yet it is limited to finite words as inputs and outputs, and to polynomial growth.
In a first part of the talk, we shall see how to use simply-typed lambda-calculus as a general mechanism for transforming trees into trees, and how it gives a positive answer to the original question. This description essentially repackages many ideas existing from the literature, in a way similar Gallot’s thesis. It also introduces a few new ones. In particular, we shall see generic results that some clearly identified classes of functional programs describing transformations from trees to trees can be effectively compiled into transducer-like models which perform only one pass on the input (i.e. higher-order tree transducers). This class of programs is also closed under composition. We shall also see how the logic MSO itself can be natively embedded in such models of computation.
In a second part, we will see how this presentation relates to recent works with Lhote, Nguy~ên and Ohlmann in which extensions of polyregular functions to maps from words to words of exponential growth are studied. This work has the particularity to be the first one in the model-theoretic side of this theory in which it is key to allow lambda-terms to be unsafe.
Cite as
Thomas Colcombet. Lambdas, Transducers and MSO (Invited Talk). In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, p. 1:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{colcombet:LIPIcs.MFCS.2025.1,
author = {Colcombet, Thomas},
title = {{Lambdas, Transducers and MSO}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {1:1--1:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.1},
URN = {urn:nbn:de:0030-drops-241087},
doi = {10.4230/LIPIcs.MFCS.2025.1},
annote = {Keywords: Lambda-calculus, automata theory, finite model theory, MSO}
}
Document
Invited Talk
Higher Connectivity in Directed Graphs (Invited Talk)
Abstract
The computation of edge-connected components in directed and undirected graphs is a well studied problem that is motivated by several applications (see, e.g., [Hiroshi Nagamochi and Toshihide Ibaraki, 2008]). Let G = (V,E) be a strongly connected directed graph with m edges and n vertices. An edge e ∈ E is a strong bridge if G ⧵ e is not strongly connected. More generally, a set of edges C ⊆ E is a cut if G ⧵ C is not strongly connected. If |C| = k then we refer to C as a k-sized cut of G. Hence, a strong bridge is a 1-sized cut of G. A digraph G is k-edge-connected if it has no (k-1)-cuts. We say that two vertices v and w are k-edge-connected, and we denote this relation by v ↔_{k} w, if there are k edge-disjoint directed paths from v to w and k edge-disjoint directed paths from w to v. (Note that a path from v to w and a path from w to v need not be edge-disjoint). By Menger’s theorem [Karl Menger, 1927], v ↔_{k} w if and only if the removal of any set of at most k-1 edges leaves v and w in the same strongly connected component. We define a k-edge-connected component of a digraph G = (V,E) as a maximal subset U ⊆ V such that u ↔_{k} v for all u, v ∈ U. The k-edge-connected components of G form a partition of V, since v ↔_{k} w is an equivalence relation [Loukas Georgiadis et al., 2016].
Connectivity-related problems are known to be much more difficult in directed graphs than in undirected graphs (see, e.g., [Harold N. Gabow, 2016; Monika Henzinger et al., 2020; Ken-Ichi Kawarabayashi and Mikkel Thorup, 2018]). Indeed, there is a fundamental difference in the structure of the cuts in the two scenarios. Specifically, it has been established more than 60 years ago [Gomory and Hu, 1961] that edge cuts in undirected graphs have a nice structure, as defined by the Gomory-Hu tree (or cut tree), which plays a special role in identifying, for any k, the k-edge-connected components of undirected graphs. Furthermore, many efficient algorithms for computing Gomory-Hu trees are available (see e.g., [Amir Abboud et al., 2021; Amir Abboud et al., 2022; Amir Abboud et al., 2023; Chen et al., 2022; Hariharan et al., 2007; Li et al., 2022]). On the contrary, in directed graphs edge cuts have a more complicated structure, and it was proved by Benczúr [Benczúr, 1995] that in this case cut trees do not even exist. It is thus not surprising that, while it is known how to compute the k-edge-connected components of undirected graphs in linear time for k ≤ 5 [Harold N. Gabow, 2000; Zvi Galil and Giuseppe F. Italiano, 1991; Loukas Georgiadis et al., 2021; John E. Hopcroft and Robert E. Tarjan, 1973; Kosinas, 2024; Wojciech Nadara et al., 2021; Hiroshi Nagamochi and Toshihide Ibaraki, 1992; Robert E. Tarjan, 1972; Yung H. Tsin, 2009], the situation is more challenging for directed graphs, where linear-time algorithms are only known for k ≤ 2 [Robert E. Tarjan, 1972; Loukas Georgiadis et al., 2020]. Also, as argued in [Loukas Georgiadis et al., 2023], there is a substantial increase in the inherent difficulty of the problem of computing k-edge-connected components in digraphs for k = 3 compared to k = 2. Indeed, for k = 2 any pair of vertices s,t that are not 2-edge-connected can be separated by only O(n) s-t min-cuts of size 1, for which we can define a total order [Giuseppe F. Italiano et al., 2012]. For k = 3, any pair of vertices s,t that are 2-edge-connected but not 3-edge-connected, can be separated by as many as O(n²) s-t min-cuts of size 2, which are also not totally ordered. This makes it difficult to explore the effect of removing each such cut of size 2 on the strong connectivity of the graph, similar to what was done for the case of k = 2 [Loukas Georgiadis et al., 2020]. Until recently, the best-known bound for computing the k-edge-connected components of a digraph, for constant k ≥ 3, was O(mn) by Nagamochi and Watanabe [Hiroshi Nagamochi and Toshimasa Watanabe, 1993]. Georgiadis et al. [Loukas Georgiadis et al., 2023] presented a randomized (Monte-Carlo) algorithm that computes the 3-edge-connected components of a digraph with m edges in Õ(m^{3/2}) time. Their algorithm involves a nontrivial extension of the framework of [Forster et al., 2020; Nanongkai et al., 2019] for deciding whether a digraph is (k+1)-edge-connected. It applies a local search procedure [Shiri Chechik et al., 2017; Forster et al., 2020] for identifying 2-in or 2-out sets, i.e., vertex sets S ⊆ V such that there are at most 2 edges from V ⧵ S to S or from S to V⧵ S. After finding such a set S, [Loukas Georgiadis et al., 2023] applies an efficient graph operation for replacing S with a gadget of small size that preserves the pairwise connectivity among the vertices of V ⧵ S. As in [Forster et al., 2020; Nanongkai et al., 2019], local search is initiated from sampled edges, but the overall scheme is more complicated to guarantee that enough 2-in sets or 2-out sets are identified that separate vertices that are not 3-edge-connected.
Recently, Georgiadis, Italiano and Kosinas [Georgiadis et al., 2024] improved significantly the bound of [Loukas Georgiadis et al., 2023] by showing how to compute the 3-edge-connected components of a digraph in linear time with a deterministic algorithm. Their algorithm differs substantially from [Loukas Georgiadis et al., 2023], as it is based on a new characterization of 2-sized cuts in digraphs, which requires new techniques and a suitable combination of the notions of 2-connectivity-light graphs [Loukas Georgiadis et al., 2023] and of maximally edge-disjoint strongly divergent spanning trees [Loukas Georgiadis and Robert E. Tarjan, 2015; Robert E. Tarjan, 1976]. In particular, Georgiadis, Italiano and Kosinas [Georgiadis et al., 2024] showed how to modify the minset-poset technique of Gabow [Harold N. Gabow, 2016], in order to find the 3-edge-connected components of a digraph with m edges in O(m) time.
In the invited talk, I will survey some of this recent work on higher connectivity on directed graphs.
Cite as
Giuseppe F. Italiano. Higher Connectivity in Directed Graphs (Invited Talk). In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 2:1-2:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{italiano:LIPIcs.MFCS.2025.2,
author = {Italiano, Giuseppe F.},
title = {{Higher Connectivity in Directed Graphs}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {2:1--2:4},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.2},
URN = {urn:nbn:de:0030-drops-241096},
doi = {10.4230/LIPIcs.MFCS.2025.2},
annote = {Keywords: Connectivity, Directed graphs, Graph algorithms}
}
Document
Invited Talk
Almost-Linear Time Algorithms for Partially Dynamic Graphs (Invited Talk)
Abstract
A partially dynamic graph is a graph that undergoes edge insertions or deletions, but not both. In this talk, I present a unifying framework that yields the first almost-optimal, almost-linear time algorithms for many well-studied problems on partially dynamic graphs [Chen-Kyng-Liu-Meierhans-Probst-Gutenberg, STOC’24; Brand-Chen-Kyng-Liu-Meierhans-Probst Gutenberg-Sachdevea, FOCS’24]. These problems include cycle detection, strongly connected components, s-t distances, transshipment, bipartite matching, maximum flow, and minimum-cost flow. We achieve this unification by solving the partially dynamic threshold minimum-cost flow problem. We solve these problems by combining a partially dynamic L1 interior point method (Brand-Liu-Sidford STOC'23) with powerful new data structures that solve fully-dynamic APSP and min-cut with sub-polynomial approximation quality and sub-polynomial update and query time.
Cite as
Rasmus Kyng. Almost-Linear Time Algorithms for Partially Dynamic Graphs (Invited Talk). In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, p. 3:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{kyng:LIPIcs.MFCS.2025.3,
author = {Kyng, Rasmus},
title = {{Almost-Linear Time Algorithms for Partially Dynamic Graphs}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {3:1--3:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.3},
URN = {urn:nbn:de:0030-drops-241109},
doi = {10.4230/LIPIcs.MFCS.2025.3},
annote = {Keywords: Graph algorithms and data strucures, continuous optimization, interior point methods}
}
Document
Invited Talk
On Graph Queries and Modal Constraints (Invited Talk)
Abstract
Some fundamental problems in database theory and knowledge representation can be viewed as instances of the query entailment problem. While query evaluation asks whether a given query holds in a specific structure, query entailment consists in determining whether the query holds in every model of a given theory that extends that structure. The input structure represents the raw data stored in a database; the theory captures contextual information such as a set of database constraints or an ontology; and the query is used to extract specific information of interest.
The recent proliferation of graph databases has brought the database and knowledge representation communities closer together, as many key problems in both fields involve the same structures - labelled graphs - and similar combinations of formalisms for theories and queries. A notable example is the combination of description logics and conjunctive regular path queries. Description logics are a family of formalisms based on fragments of first order logic, akin to modal logics. They are among the most prominent ontology languages and they are capable of expressing most kinds of constraints relevant in graph databases. Conjunctive regular path queries extend conjunctive queries (primitive positive first-order formulas) by allowing regular expressions over binary predicates to be used as atoms. They form the core of practical query languages employed in graph database systems and the Semantic Web.
What distinguishes the two fields is the approach to infinity: knowledge representation embraces infinite models, whereas database theory focuses on finite models. Although many cases of the entailment problem have long been solved over unrestricted (finite or infinite) models, their finite-model counterparts have only recently seen progress, and many questions remain open.
Cite as
Filip Murlak. On Graph Queries and Modal Constraints (Invited Talk). In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, p. 4:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{murlak:LIPIcs.MFCS.2025.4,
author = {Murlak, Filip},
title = {{On Graph Queries and Modal Constraints}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {4:1--4:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.4},
URN = {urn:nbn:de:0030-drops-241117},
doi = {10.4230/LIPIcs.MFCS.2025.4},
annote = {Keywords: conjunctive regular path queries, query entailment, query containment, graph databases, database schemas, integrity constraints, description logics, finite-model reasoning, ontology-mediated query answering, static analysis}
}
Document
Invited Talk
On Synthesis of Distributed Monitors (Invited Talk)
Abstract
This talk addresses the synthesis problem of distributed monitors for concurrency properties.
Cite as
Anca Muscholl. On Synthesis of Distributed Monitors (Invited Talk). In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 5:1-5:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{muscholl:LIPIcs.MFCS.2025.5,
author = {Muscholl, Anca},
title = {{On Synthesis of Distributed Monitors}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {5:1--5:3},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.5},
URN = {urn:nbn:de:0030-drops-241126},
doi = {10.4230/LIPIcs.MFCS.2025.5},
annote = {Keywords: Distributed synthesis, monitoring}
}
Document
Catalytic Computing and Register Programs Beyond Log-Depth
Abstract
In a seminal work, Buhrman et al. (STOC 2014) defined the class CSPACE(s,c) of problems solvable in space s with an additional catalytic tape of size c, which is a tape whose initial content must be restored at the end of the computation. They showed that uniform TC¹ circuits are computable in catalytic logspace, i.e., CL = CSPACE(O(log{n}), 2^{O(log{n})}), thus giving strong evidence that catalytic space gives L strict additional power. Their study focuses on an arithmetic model called register programs, which has been a focal point in development since then.
Understanding CL remains a major open problem, as TC¹ remains the most powerful containment to date. In this work, we study the power of catalytic space and register programs to compute circuits of larger depth. Using register programs, we show that for every ε > 0,
SAC² ⊆ CSPACE (O((log²n)/(log log n)), 2^{O(log^{1+ε} n)}) .
On the other hand, we know that SAC² ⊆ TC² ⊆ CSPACE(O(log²{n}) , 2^{O(log{n})}). Our result thus shows an O(log log n) factor improvement on the free space needed to compute SAC², at the expense of a nearly-polynomial-sized catalytic tape.
We also exhibit non-trivial register programs for matrix powering, which is a further step towards showing NC² ⊆ CL.
Cite as
Yaroslav Alekseev, Yuval Filmus, Ian Mertz, Alexander Smal, and Antoine Vinciguerra. Catalytic Computing and Register Programs Beyond Log-Depth. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{alekseev_et_al:LIPIcs.MFCS.2025.6,
author = {Alekseev, Yaroslav and Filmus, Yuval and Mertz, Ian and Smal, Alexander and Vinciguerra, Antoine},
title = {{Catalytic Computing and Register Programs Beyond Log-Depth}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {6:1--6:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.6},
URN = {urn:nbn:de:0030-drops-241136},
doi = {10.4230/LIPIcs.MFCS.2025.6},
annote = {Keywords: catalytic computing, circuit classes, polynomial method}
}
Document
Approximating Prize-Collecting Variants of TSP
Abstract
We present an approximation algorithm for the Prize-collecting Ordered Traveling Salesman Problem (PCOTSP), which simultaneously generalizes the Prize-collecting TSP and the Ordered TSP. The Prize-collecting TSP is well-studied and has a long history, with the current best approximation factor slightly below 1.6, shown by Blauth, Klein and Nägele [IPCO 2024]. The best approximation ratio for Ordered TSP is 3/2+1/e, presented by Böhm, Friggstad, Mömke, Spoerhase [SODA 2025] and Armbruster, Mnich, Nägele [Approx 2024]. The former also present a factor 2.2131 approximation algorithm for Multi-Path-TSP.
We present a 2.097-approximation algorithm for PCOTSP, which is, to the best of our knowledge, the first result for this problem. Key ideas in our approach are to sample a set of trees and then to probabilistically pick up some vertices, and to use the pruning ideas of Blauth, Klein, Nägele [IPCO 2024] on the sampled vertices. While the sampling probability of vertices for our problem is lower than for PCTSP, intuitively leaving less spare penalty to spend, we leverage the cycle structure induced by the sampled trees together with a simple combinatorial algorithm to bring the approximation factor below 2.1.
Our techniques extend to Prize-collecting Multi-Path TSP, building on results from Böhm, Friggstad, Mömke, Spoerhase [SODA 2025], leading to a 2.41-approximation.
Cite as
Morteza Alimi, Tobias Mömke, and Michael Ruderer. Approximating Prize-Collecting Variants of TSP. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{alimi_et_al:LIPIcs.MFCS.2025.7,
author = {Alimi, Morteza and M\"{o}mke, Tobias and Ruderer, Michael},
title = {{Approximating Prize-Collecting Variants of TSP}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {7:1--7:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.7},
URN = {urn:nbn:de:0030-drops-241141},
doi = {10.4230/LIPIcs.MFCS.2025.7},
annote = {Keywords: Approximation Algorithms, TSP}
}
Document
Dynamic Membership for Regular Tree Languages
Abstract
We study the dynamic membership problem for regular tree languages under relabeling updates: we fix an alphabet Σ and a regular tree language L over Σ (expressed, e.g., as a tree automaton), we are given a tree T with labels in Σ, and we must maintain the information of whether the tree T belongs to L while handling relabeling updates that change the labels of individual nodes in T.
Our first contribution is to show that this problem admits an O(log n / log log n) algorithm for any fixed regular tree language, improving over known O(log n) algorithms. This generalizes the known O(log n / log log n) upper bound over words, and it matches the lower bound of Ω(log n / log log n) from dynamic membership to some word languages and from the existential marked ancestor problem.
Our second contribution is to introduce a class of regular languages, dubbed almost-commutative tree languages, and show that dynamic membership to such languages under relabeling updates can be decided in constant time per update. Almost-commutative languages generalize both commutative languages and finite languages: they are the analogue for trees of the ZG languages enjoying constant-time dynamic membership over words. Our main technical contribution is to show that this class is conditionally optimal when we assume that the alphabet features a neutral letter, i.e., a letter that has no effect on membership to the language. More precisely, we show that any regular tree language with a neutral letter which is not almost-commutative cannot be maintained in constant time under the assumption that the prefix-U1 problem from [Antoine Amarilli et al., 2021] also does not admit a constant-time algorithm.
Cite as
Antoine Amarilli, Corentin Barloy, Louis Jachiet, and Charles Paperman. Dynamic Membership for Regular Tree Languages. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{amarilli_et_al:LIPIcs.MFCS.2025.8,
author = {Amarilli, Antoine and Barloy, Corentin and Jachiet, Louis and Paperman, Charles},
title = {{Dynamic Membership for Regular Tree Languages}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {8:1--8:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.8},
URN = {urn:nbn:de:0030-drops-241155},
doi = {10.4230/LIPIcs.MFCS.2025.8},
annote = {Keywords: automaton, dynamic membership, incremental maintenance, forest algebra}
}
Document
Linear Time Subsequence and Supersequence Regex Matching
Abstract
It is well-known that checking whether a given string w matches a given regular expression r can be done in quadratic time O(|w|⋅ |r|) and that this cannot be improved to a truly subquadratic running time of O((|w|⋅ |r|)^{1-ε}) assuming the strong exponential time hypothesis (SETH). We study a different matching paradigm where we ask instead whether w has a subsequence that matches r, and show that regex matching in this sense can be solved in linear time O(|w| + |r|). Further, the same holds if we ask for a supersequence. We show that the quantitative variants where we want to compute a longest or shortest subsequence or supersequence of w that matches r can be solved in O(|w|⋅ |r|), i. e., asymptotically no worse than classical regex matching; and we show that O(|w| + |r|) is conditionally not possible for these problems. We also investigate these questions with respect to other natural string relations like the infix, prefix, left-extension or extension relation instead of the subsequence and supersequence relation. We further study the complexity of the universal problem where we ask if all subsequences (or supersequences, infixes, prefixes, left-extensions or extensions) of an input string satisfy a given regular expression.
Cite as
Antoine Amarilli, Florin Manea, Tina Ringleb, and Markus L. Schmid. Linear Time Subsequence and Supersequence Regex Matching. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{amarilli_et_al:LIPIcs.MFCS.2025.9,
author = {Amarilli, Antoine and Manea, Florin and Ringleb, Tina and Schmid, Markus L.},
title = {{Linear Time Subsequence and Supersequence Regex Matching}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {9:1--9:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.9},
URN = {urn:nbn:de:0030-drops-241162},
doi = {10.4230/LIPIcs.MFCS.2025.9},
annote = {Keywords: subsequence, supersequence, regular language, regular expression, automata}
}
Document
Characterizing Small Circuit Classes from FAC⁰ to FAC¹ via Discrete Ordinary Differential Equations
Abstract
In this paper, we provide a uniform framework for investigating small circuit classes and bounds through the lens of ordinary differential equations (ODEs). Following an approach recently introduced to capture the class of polynomial-time computable functions via ODE-based recursion schemas and later applied to the context of functions computed by unbounded fan-in circuits of constant depth (FAC⁰), we study multiple relevant small circuit classes. In particular, we show that natural restrictions on linearity and derivation along functions with specific growth rate correspond to kinds of functions that can be proved to be in various classes, ranging from FAC⁰ to FAC¹. This reveals an intriguing link between constraints over linear-length ODEs and circuit computation, providing new tools to tackle the complex challenge of establishing bounds for classes in the circuit hierarchies and possibly enhancing our understanding of the role of counters in this setting. Additionally, we establish several completeness results, in particular obtaining the first ODE-based characterizations for the classes of functions computable in constant depth with unbounded fan-in and Mod 2 gates (FACC[2]) and in logarithmic depth with bounded fan-in Boolean gates (FNC¹).
Cite as
Melissa Antonelli, Arnaud Durand, and Juha Kontinen. Characterizing Small Circuit Classes from FAC⁰ to FAC¹ via Discrete Ordinary Differential Equations. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 10:1-10:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{antonelli_et_al:LIPIcs.MFCS.2025.10,
author = {Antonelli, Melissa and Durand, Arnaud and Kontinen, Juha},
title = {{Characterizing Small Circuit Classes from FAC⁰ to FAC¹ via Discrete Ordinary Differential Equations}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {10:1--10:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.10},
URN = {urn:nbn:de:0030-drops-241170},
doi = {10.4230/LIPIcs.MFCS.2025.10},
annote = {Keywords: Implicit computational complexity, parallel computation, function algebras, ordinary differential equations, circuit complexity}
}
Document
Universality Frontier for Asynchronous Cellular Automata
Abstract
In this work, we investigate computational aspects of asynchronous cellular automata (ACAs), a modification of cellular automata in which cells update independently, following an asynchronous update schedule. We introduce flip automata networks (FANs), a simple modification of automata networks that remain robust under any asynchronous updating order. By using FANs as a middleman, we show that asynchronous automata can efficiently simulate their synchronous counterparts with a linear memory overhead, which improves upon the previously established quadratic bound. Additionally, we address the universality gap for (a)synchronous cellular automata-the boundary separating universal and non-universal automata, which is still not fully understood. We tighten this boundary by proving that all one-way asynchronous automata lack universal computational power. Conversely, we establish the existence of a universal asynchronous 6-state first-neighbor automaton in one dimension and a 3-state von Neumann automaton in two dimensions, which represent the smallest known universal constructions to date.
Cite as
Ivan Baburin, Matthew Cook, Florian Grötschla, Andreas Plesner, and Roger Wattenhofer. Universality Frontier for Asynchronous Cellular Automata. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 11:1-11:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{baburin_et_al:LIPIcs.MFCS.2025.11,
author = {Baburin, Ivan and Cook, Matthew and Gr\"{o}tschla, Florian and Plesner, Andreas and Wattenhofer, Roger},
title = {{Universality Frontier for Asynchronous Cellular Automata}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {11:1--11:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.11},
URN = {urn:nbn:de:0030-drops-241182},
doi = {10.4230/LIPIcs.MFCS.2025.11},
annote = {Keywords: Universality, Asynchronous Cellular Automata, Automata Networks}
}
Document
Online Knapsack Problems with Estimates
Abstract
Imagine you are a computer scientist who enjoys attending conferences or workshops within the year. Sadly, your travel budget is limited, so you must select a subset of events you can travel to. When you are aware of all possible events and their costs at the beginning of the year, you can select the subset of the possible events that maximizes your happiness and is within your budget. On the other hand, if you are blind about the options, you will likely have a hard time when trying to decide if you want to register somewhere or not, and will likely regret decisions you made in the future. These scenarios can be modeled by knapsack variants, either by an offline or an online problem. However, both scenarios are somewhat unrealistic: Usually, you will not know the exact costs of each workshop at the beginning of the year. The online version, however, is too pessimistic, as you might already know which options there are and how much they cost roughly. At some point, you have to decide whether to register for some workshop, but then you are aware of the conference fee and the flight and hotel prices.
We model this problem within the setting of online knapsack problems with estimates: in the beginning, you receive a list of potential items with their estimated size as well as the accuracy of the estimates. Then, the items are revealed one by one in an online fashion with their actual size, and you need to decide whether to take one or not. In this article, we show a best-possible algorithm for each estimate accuracy δ (i.e., when each actual item size can deviate by ± δ from the announced size) for both the simple knapsack (also known as subset sum problem) and the simple knapsack with removability.
Cite as
Jakub Balabán, Matthias Gehnen, Henri Lotze, Finn Seesemann, and Moritz Stocker. Online Knapsack Problems with Estimates. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 12:1-12:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{balaban_et_al:LIPIcs.MFCS.2025.12,
author = {Balab\'{a}n, Jakub and Gehnen, Matthias and Lotze, Henri and Seesemann, Finn and Stocker, Moritz},
title = {{Online Knapsack Problems with Estimates}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {12:1--12:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.12},
URN = {urn:nbn:de:0030-drops-241190},
doi = {10.4230/LIPIcs.MFCS.2025.12},
annote = {Keywords: Knapsack, Online Knapsack, Removability, Estimate, Prediction}
}
Document
Solving Partial Dominating Set and Related Problems Using Twin-Width
Abstract
Partial vertex cover and partial dominating set are two well-investigated optimization problems. While they are W[1]-hard on general graphs, they have been shown to be fixed-parameter tractable on many sparse graph classes, including nowhere-dense classes. In this paper, we demonstrate that these problems are also fixed-parameter tractable with respect to the twin-width of a graph. Indeed, we establish a more general result: every graph property that can be expressed by a logical formula of the form ϕ≡∃ x₁⋯ ∃ x_k ∑_{α ∈ I} #y ψ_α(x₁,…,x_k,y) ≥ t, where ψ_α is a quantifier-free formula for each α ∈ I, t is an arbitrary number, and #y is a counting quantifier, can be evaluated in time f(d,k)n, where n is the number of vertices and d is the width of a contraction sequence that is part of the input. In addition to the aforementioned problems, this includes also connected partial dominating set and independent partial dominating set.
Cite as
Jakub Balabán, Daniel Mock, and Peter Rossmanith. Solving Partial Dominating Set and Related Problems Using Twin-Width. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 13:1-13:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{balaban_et_al:LIPIcs.MFCS.2025.13,
author = {Balab\'{a}n, Jakub and Mock, Daniel and Rossmanith, Peter},
title = {{Solving Partial Dominating Set and Related Problems Using Twin-Width}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {13:1--13:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.13},
URN = {urn:nbn:de:0030-drops-241203},
doi = {10.4230/LIPIcs.MFCS.2025.13},
annote = {Keywords: Partial Dominating Set, Partial Vertex Cover, meta-algorithm, counting logic, twin-width}
}
Document
Register Automata with Permutations
Abstract
We propose Permutation Deterministic Register Automata (pDRAs), a deterministic register automaton model where we allow permutations of registers in transitions. The model enables minimal canonical representations and pDRAs can be tested for equivalence in polynomial time. The complexity of minimization is between GI (the complexity of graph isomorphism) and NP. We then introduce a subclass of pDRAs, called register automata with fixed permutation policy, where the register permutation discipline is stipulated globally. This class generalizes the model proposed by Benedikt, Ley and Puppis in 2010, and we show that it also admits minimal and canonical representations, based on a finite-index word equivalence relation. As an application, we show that for any regular data language L, the minimal register automaton with fixed permutation policy recognizing L can be actively learned in polynomial time using oracles for membership, equivalence and data-memorability queries. We show that all the oracles can be implemented in polynomial time, and so this yields a polynomial time minimization algorithm.
Cite as
Mrudula Balachander, Emmanuel Filiot, Raffaella Gentilini, and Nikos Tzevelekos. Register Automata with Permutations. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{balachander_et_al:LIPIcs.MFCS.2025.14,
author = {Balachander, Mrudula and Filiot, Emmanuel and Gentilini, Raffaella and Tzevelekos, Nikos},
title = {{Register Automata with Permutations}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {14:1--14:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.14},
URN = {urn:nbn:de:0030-drops-241219},
doi = {10.4230/LIPIcs.MFCS.2025.14},
annote = {Keywords: Register automata, data words, equivalence, minimization, active learning}
}
Document
Symmetry Classes of Hamiltonian Cycles
Abstract
We initiate the study of Hamiltonian cycles up to symmetries of the underlying graph. Our focus lies on the extremal case of Hamiltonian-transitive graphs, i.e., Hamiltonian graphs where, for every pair of Hamiltonian cycles, there is a graph automorphism mapping one cycle to the other. This generalizes the extensively studied uniquely Hamiltonian graphs. In this paper, we show that Cayley graphs of abelian groups are not Hamiltonian-transitive (under some mild conditions and some non-surprising exceptions), i.e., they contain at least two structurally different Hamiltonian cycles. To show this, we reduce Hamiltonian-transitivity to properties of the prime factors of a Cartesian product decomposition, which we believe is interesting in its own right. We complement our results by constructing infinite families of regular Hamiltonian-transitive graphs and take a look at the opposite extremal case by constructing a family with many different Hamiltonian cycles up to symmetry.
Cite as
Júlia Baligács, Sofia Brenner, Annette Lutz, and Lena Volk. Symmetry Classes of Hamiltonian Cycles. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{baligacs_et_al:LIPIcs.MFCS.2025.15,
author = {Balig\'{a}cs, J\'{u}lia and Brenner, Sofia and Lutz, Annette and Volk, Lena},
title = {{Symmetry Classes of Hamiltonian Cycles}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {15:1--15:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.15},
URN = {urn:nbn:de:0030-drops-241221},
doi = {10.4230/LIPIcs.MFCS.2025.15},
annote = {Keywords: Hamiltonian cycles, graph automorphisms, Cayley graphs, abelian groups, Cartesian product of graphs}
}
Document
Isometric-Universal Graphs for Trees
Abstract
We consider the problem of finding the smallest graph that contains two input trees each with at most n vertices preserving their distances. In other words, we look for an isometric-universal graph with the minimum number of vertices for two given trees. We prove that this problem can be solved in time O(n^{5/2}log{n}). We extend this result to forests instead of trees, and propose an algorithm with running time O(n^{7/2}log{n}). As a key ingredient, we show that a smallest isometric-universal graph of two trees essentially is a tree. Furthermore, we prove that these results cannot be extended. Firstly, we show that deciding whether there exists an isometric-universal graph with t vertices for three forests is NP-complete. Secondly, we show that any smallest isometric-universal graph cannot be a tree for some families of three trees. This latter result has implications for greedy strategies solving the smallest isometric-universal graph problem.
Cite as
Edgar Baucher, François Dross, and Cyril Gavoille. Isometric-Universal Graphs for Trees. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 16:1-16:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{baucher_et_al:LIPIcs.MFCS.2025.16,
author = {Baucher, Edgar and Dross, Fran\c{c}ois and Gavoille, Cyril},
title = {{Isometric-Universal Graphs for Trees}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {16:1--16:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.16},
URN = {urn:nbn:de:0030-drops-241237},
doi = {10.4230/LIPIcs.MFCS.2025.16},
annote = {Keywords: tree, forest, isometric subgraph, universal graph, distance-preserving}
}
Document
Sensitivity and Query Complexity Under Uncertainty
Abstract
In this paper, we study the query complexity of Boolean functions in the presence of uncertainty, motivated by parallel computation with an unlimited number of processors where inputs are allowed to be unknown. We allow each query to produce three results: zero, one, or unknown. The output could also be: zero, one, or unknown, with the constraint that we should output "unknown" only when we cannot determine the answer from the revealed input bits. Such an extension of a Boolean function is called its hazard-free extension.
- We prove an analogue of Huang’s celebrated sensitivity theorem [Annals of Mathematics, 2019] in our model of query complexity with uncertainty.
- We show that the deterministic query complexity of the hazard-free extension of a Boolean function is at most quadratic in its randomized query complexity and quartic in its quantum query complexity, improving upon the best-known bounds in the Boolean world.
- We exhibit an exponential gap between the smallest depth (size) of decision trees computing a Boolean function, and those computing its hazard-free extension.
- We present general methods to convert decision trees for Boolean functions to those for their hazard-free counterparts, and show optimality of this construction. We also parameterize this result by the maximum number of unknown values in the input.
- We show lower bounds on size complexity of decision trees for hazard-free extensions of Boolean functions in terms of the number of prime implicants and prime implicates of the underlying Boolean function.
Cite as
Deepu Benson, Balagopal Komarath, Nikhil Mande, Sai Soumya Nalli, Jayalal Sarma, and Karteek Sreenivasaiah. Sensitivity and Query Complexity Under Uncertainty. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 17:1-17:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{benson_et_al:LIPIcs.MFCS.2025.17,
author = {Benson, Deepu and Komarath, Balagopal and Mande, Nikhil and Nalli, Sai Soumya and Sarma, Jayalal and Sreenivasaiah, Karteek},
title = {{Sensitivity and Query Complexity Under Uncertainty}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {17:1--17:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.17},
URN = {urn:nbn:de:0030-drops-241240},
doi = {10.4230/LIPIcs.MFCS.2025.17},
annote = {Keywords: CREW-PRAM, query complexity, decision trees, sensitivity, hazard-free extensions}
}
Document
Supercritical Size-Width Tree-Like Resolution Trade-Offs for Graph Isomorphism
Abstract
We study the refutation complexity of graph isomorphism in the tree-like resolution calculus. Torán and Wörz [Jacobo Torán and Florian Wörz, 2023] showed that there is a resolution refutation of narrow width k for two graphs if and only if they can be distinguished in (k+1)-variable first-order logic (FO^{k+1}). While DAG-like narrow width k resolution refutations have size at most n^k, tree-like refutations may be much larger. We show that there are graphs of order n, whose isomorphism can be refuted in narrow width k but only in tree-like size 2^{Ω(n^{k/2})}. This is a supercritical trade-off where bounding one parameter (the narrow width) causes the other parameter (the size) to grow above its worst case. The size lower bound is super-exponential in the formula size and improves a related supercritical trade-off by Razborov [Alexander A. Razborov, 2016]. To prove our result, we develop a new variant of the k-pebble EF-game for FO^k to reason about tree-like refutation size in a similar way as the Prover-Delayer games in proof complexity. We analyze this game on the compressed CFI graphs introduced by Grohe, Lichter, Neuen, and Schweitzer [Martin Grohe et al., 2023]. Using a recent improved robust compressed CFI construction of de Rezende, Fleming, Janett, Nordström, and Pang [Susanna F. de Rezende et al., 2024], we obtain a similar bound for width k (instead of the stronger but less common narrow width) and make the result more robust.
Cite as
Christoph Berkholz, Moritz Lichter, and Harry Vinall-Smeeth. Supercritical Size-Width Tree-Like Resolution Trade-Offs for Graph Isomorphism. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 18:1-18:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{berkholz_et_al:LIPIcs.MFCS.2025.18,
author = {Berkholz, Christoph and Lichter, Moritz and Vinall-Smeeth, Harry},
title = {{Supercritical Size-Width Tree-Like Resolution Trade-Offs for Graph Isomorphism}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {18:1--18:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.18},
URN = {urn:nbn:de:0030-drops-241253},
doi = {10.4230/LIPIcs.MFCS.2025.18},
annote = {Keywords: Proof complexity, Resolution, Width, Tree-like size, Supercritical trade-off, Lower bound, Finite model theory, CFI graphs}
}
Document
Monotone Bounded-Depth Complexity of Homomorphism Polynomials
Abstract
For every fixed graph H, it is known that homomorphism counts from H and colorful H-subgraph counts can be determined in O(n^{t+1}) time on n-vertex input graphs G, where t is the treewidth of H. On the other hand, a running time of n^{o(t / log t)} would refute the exponential-time hypothesis. Komarath, Pandey, and Rahul (Algorithmica, 2023) studied algebraic variants of these counting problems, i.e., homomorphism and subgraph polynomials for fixed graphs H. These polynomials are weighted sums over the objects counted above, where each object is weighted by the product of variables corresponding to edges contained in the object. As shown by Komarath et al., the monotone circuit complexity of the homomorphism polynomial for H is Θ(n^{tw(H)+1}).
In this paper, we characterize the power of monotone bounded-depth circuits for homomorphism and colorful subgraph polynomials. This leads us to discover a natural hierarchy of graph parameters tw_Δ(H), for fixed Δ ∈ ℕ, which capture the width of tree-decompositions for H when the underlying tree is required to have depth at most Δ. We prove that monotone circuits of product-depth Δ computing the homomorphism polynomial for H require size Θ(n^{tw_Δ(H^{†})+1}), where H^{†} is the graph obtained from H by removing all degree-1 vertices. This allows us to derive an optimal depth hierarchy theorem for monotone bounded-depth circuits through graph-theoretic arguments.
Cite as
C.S. Bhargav, Shiteng Chen, Radu Curticapean, and Prateek Dwivedi. Monotone Bounded-Depth Complexity of Homomorphism Polynomials. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 19:1-19:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{bhargav_et_al:LIPIcs.MFCS.2025.19,
author = {Bhargav, C.S. and Chen, Shiteng and Curticapean, Radu and Dwivedi, Prateek},
title = {{Monotone Bounded-Depth Complexity of Homomorphism Polynomials}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {19:1--19:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.19},
URN = {urn:nbn:de:0030-drops-241269},
doi = {10.4230/LIPIcs.MFCS.2025.19},
annote = {Keywords: algebraic complexity, homomorphisms, monotone circuit complexity, bounded-depth circuits, treewidth, pathwidth}
}
Document
Kernelization in Almost Linear Time for Clustering into Bounded Vertex Cover Components
Abstract
Motivated by the growing interest in graph clustering and the framework proposed during the Dagstuhl Seminar 23331, we consider a natural specialization of this general approach (as also suggested during the seminar). The seminar introduced a broad perspective on clustering, where the goal is to partition a graph into connected components (or "clusters") that satisfy simple structural integrity constraints - not necessarily limited to cliques.
In our work, we focus on the case where each cluster is required to have bounded vertex cover number. Specifically, a connected component C satisfies this condition if there exists a set S ⊆ V(C) with |S| ≤ d such that C - S is an independent set. We study this within the framework of the {Vertex Deletion to d-Vertex Cover Components} ({Vertex Deletion to d-VCC}) problem: given a graph G and an integer k, the task is to determine whether there exists a vertex set S ⊆ V(G) of size at most k such that every connected component of G - S has vertex cover number at most d. We also examine the edge-deletion variant, {Edge Deletion to d-Vertex Cover Components} ({Edge Deletion to d-VCC}), where the goal is to delete at most k edges so that each connected component of the resulting graph has vertex cover number at most d. We obtain following results.
1) {Vertex Deletion to d-VCC} admits a kernel with {𝒪}(d⁶k³) vertices and {𝒪}(d⁹k⁴) edges.
2) {Edge Deletion to d-VCC}, admits a kernel with {𝒪}(d⁴k) vertices and {𝒪}(d⁵k) edges. Both of our kernelization algorithms run in time 𝒪(1.253^d ⋅ (kd)^{𝒪(1)} ⋅ n log n). It is important to note that, unless the Exponential Time Hypothesis (ETH) fails, the dependence on d cannot be improved to 2^{o(d)}, as the case k = 0 reduces to solving the classical Vertex Cover problem, which is known to require 2^{Ω(d)} time under ETH.
A key ingredient in our kernelization algorithms is a structural result about the hereditary graph class 𝒢_d, consisting of graphs in which every connected component has vertex cover number at most d. We show that 𝒢_d admits a finite obstruction set (with respect to the induced subgraph relation) of size 2^{𝒪(d²)}, where each obstruction graph has at most 3d + 2 vertices. This combinatorial result may be of independent interest.
Cite as
Sriram Bhyravarapu, Pritesh Kumar, Madhumita Kundu, Shivesh K. Roy, Sahiba, and Saket Saurabh. Kernelization in Almost Linear Time for Clustering into Bounded Vertex Cover Components. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{bhyravarapu_et_al:LIPIcs.MFCS.2025.20,
author = {Bhyravarapu, Sriram and Kumar, Pritesh and Kundu, Madhumita and Roy, Shivesh K. and Sahiba and Saurabh, Saket},
title = {{Kernelization in Almost Linear Time for Clustering into Bounded Vertex Cover Components}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {20:1--20:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.20},
URN = {urn:nbn:de:0030-drops-241276},
doi = {10.4230/LIPIcs.MFCS.2025.20},
annote = {Keywords: Parameterized complexity, Polynomial Kernels, Vertex Cover, Finite Forbidden Characterization}
}
Document
On the Performance of Mildly Greedy Players in k-Coloring Games
Abstract
We study the performance of mildly greedy players in k-coloring games, a relevant subclass of anti-coordination games. A mildly greedy player is a selfish agent who is willing to deviate from a certain strategy profile only if her payoff improves by a factor of more than ε, for some given ε ≥ 0. In presence of mildly greedy players, stability is captured by the concept of (1+ε)-approximate Nash equilibrium. In this paper, we first show that, for any k-coloring game, the (1+ε)-approximate price of anarchy, i.e., the price of anarchy of (1+ε)-approximate pure Nash equilibria, is at least (k-1)/((k-1)ε +k), and that this bound is tight for any ε ≥ 0. Then, we evaluate the approximation ratio of the solutions achieved after a (1 + ϵ)-approximate one-round walk starting from any initial strategy profile, where a (1 + ϵ)-approximate one-round walk is a sequence of (1 + ε)-approximate best-responses, one for each player. We provide a lower bound of min{(k-2)/k, (k-1)/((k-1)ε+k)} on this ratio, for any ε ≥ 0 and k ≥ 5; for the cases of k = 3 and k = 4, we give finer bounds depending on ε. Our work generalizes the results known for cut games, the special case of k-coloring games restricted to k = 2.
Cite as
Vittorio Bilò, Andrea D'Ascenzo, Mattia D'Emidio, and Giuseppe F. Italiano. On the Performance of Mildly Greedy Players in k-Coloring Games. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 21:1-21:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{bilo_et_al:LIPIcs.MFCS.2025.21,
author = {Bil\`{o}, Vittorio and D'Ascenzo, Andrea and D'Emidio, Mattia and Italiano, Giuseppe F.},
title = {{On the Performance of Mildly Greedy Players in k-Coloring Games}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {21:1--21:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.21},
URN = {urn:nbn:de:0030-drops-241287},
doi = {10.4230/LIPIcs.MFCS.2025.21},
annote = {Keywords: Coloring games, (Approximate) Nash Equilibria, Price of Anarchy}
}
Document
On the Reachability Problem for Two-Dimensional Branching VASS
Abstract
Vectors addition systems with states (VASS), or equivalently Petri nets, are arguably one of the most studied formalisms for the modeling and analysis of concurrent systems. A central decision problem for VASS is reachability: whether there exists a run from an initial configuration to a final one. This problem has been known to be decidable for over forty years, and its complexity has recently been precisely characterized. Our work concerns the reachability problem for BVASS, a branching generalization of VASS. In dimension one, the exact complexity of this problem is known. In this paper, we prove that the reachability problem for 2-dimensional BVASS is decidable. In fact, we even show that the reachability set admits a computable semilinear presentation. The decidability status of the reachability problem for BVASS remains open in higher dimensions.
Cite as
Clotilde Bizière, Thibault Hilaire, Jérôme Leroux, and Grégoire Sutre. On the Reachability Problem for Two-Dimensional Branching VASS. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 22:1-22:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{biziere_et_al:LIPIcs.MFCS.2025.22,
author = {Bizi\`{e}re, Clotilde and Hilaire, Thibault and Leroux, J\'{e}r\^{o}me and Sutre, Gr\'{e}goire},
title = {{On the Reachability Problem for Two-Dimensional Branching VASS}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {22:1--22:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.22},
URN = {urn:nbn:de:0030-drops-241294},
doi = {10.4230/LIPIcs.MFCS.2025.22},
annote = {Keywords: Vector addition systems, Reachability problem, Semilinear sets, Verification}
}
Document
Which Graph Motif Parameters Count?
Abstract
For a fixed graph H, the function #Ind(H → ⋆) maps graphs G to the count of induced H-copies in G; this function obviously "counts something" in that it has a combinatorial interpretation. Linear combinations of such functions are called graph motif parameters and have recently received significant attention in counting complexity after a seminal paper by Curticapean, Dell and Marx (STOC'17). We show that, among linear combinations of functions #Ind(H → ⋆) involving only graphs H without isolated vertices, precisely those with positive integer coefficients maintain a combinatorial interpretation. It is important to note that graph motif parameters can be nonnegative for all inputs G, even when some coefficients are negative.
Formally, we show that evaluating any graph motif parameter with a negative coefficient is impossible in an oracle variant of #P, where an implicit graph is accessed by oracle queries. Our proof follows the classification of the relativizing closure properties of #P by Hertrampf, Vollmer, and Wagner (SCT'95) and the framework developed by Ikenmeyer and Pak (STOC'22), but our application of the required Ramsey theorem turns out to be more subtle, as graphs do not have the required Ramsey property.
Our techniques generalize from graphs to relational structures, including colored graphs. Vastly generalizing this, we introduce motif parameters over categories that count occurrences of sub-objects in the category. We then prove a general dichotomy theorem that characterizes which such parameters have a combinatorial interpretation. Using known results in Ramsey theory for categories, we obtain a dichotomy for motif parameters of finite vector spaces as well as parameter sets.
Cite as
Markus Bläser, Radu Curticapean, Julian Dörfler, and Christian Ikenmeyer. Which Graph Motif Parameters Count?. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{blaser_et_al:LIPIcs.MFCS.2025.23,
author = {Bl\"{a}ser, Markus and Curticapean, Radu and D\"{o}rfler, Julian and Ikenmeyer, Christian},
title = {{Which Graph Motif Parameters Count?}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {23:1--23:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.23},
URN = {urn:nbn:de:0030-drops-241307},
doi = {10.4230/LIPIcs.MFCS.2025.23},
annote = {Keywords: Graph motif parameters, Combinatorics, Combinatorial Interpretability}
}
Document
Temporal Valued Constraint Satisfaction Problems
Abstract
We study the computational complexity of the valued constraint satisfaction problem (VCSP) for every valued structure over ℚ that is preserved by all order-preserving bijections. Such VCSPs will be called temporal, in analogy to the (classical) constraint satisfaction problem: a relational structure is preserved by all order-preserving bijections if and only if all its relations have a first-order definition in (ℚ; <), and the CSPs for such structures are called temporal CSPs. Many optimization problems that have been studied intensively in the literature can be phrased as a temporal VCSP. We prove that a temporal VCSP is in P, or NP-complete. Our analysis uses the concept of fractional polymorphisms. This is the first dichotomy result for VCSPs over infinite domains which is complete in the sense that it treats all valued structures that contain a given automorphism group.
Cite as
Manuel Bodirsky, Édouard Bonnet, and Žaneta Semanišinová. Temporal Valued Constraint Satisfaction Problems. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 24:1-24:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{bodirsky_et_al:LIPIcs.MFCS.2025.24,
author = {Bodirsky, Manuel and Bonnet, \'{E}douard and Semani\v{s}inov\'{a}, \v{Z}aneta},
title = {{Temporal Valued Constraint Satisfaction Problems}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {24:1--24:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.24},
URN = {urn:nbn:de:0030-drops-241311},
doi = {10.4230/LIPIcs.MFCS.2025.24},
annote = {Keywords: Constraint Satisfaction Problems, valued CSPs, temporal CSPs, fractional polymorphisms, complexity dichotomy, min CSPs}
}
Document
Polynomial-Time Tractable Problems over the p-Adic Numbers
Abstract
We study the computational complexity of fundamental problems over the p-adic numbers {ℚ}_p and the p-adic integers {ℤ}_p. Guépin, Haase, and Worrell [Florent Guépin et al., 2019] proved that checking satisfiability of systems of linear equations combined with valuation constraints of the form v_p(x) = c for p ≥ 5 is NP-complete (both over {ℤ}_p and over {ℚ}_p), and left the cases p = 2 and p = 3 open. We solve their problem by showing that the problem is NP-complete for {ℤ}₃ and for {ℚ}₃, but that it is in P for {ℤ}₂ and for {ℚ}₂. We also present different polynomial-time algorithms for solvability of systems of linear equations in {ℚ}_p with either constraints of the form v_p(x) ≤ c or of the form v_p(x) ≥ c for c ∈ {ℤ}. Finally, we show how our algorithms can be used to decide in polynomial time the satisfiability of systems of (strict and non-strict) linear inequalities over {ℚ} together with valuation constraints v_p(x) ≥ c for several different prime numbers p simultaneously.
Cite as
Manuel Bodirsky and Arno Fehm. Polynomial-Time Tractable Problems over the p-Adic Numbers. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{bodirsky_et_al:LIPIcs.MFCS.2025.25,
author = {Bodirsky, Manuel and Fehm, Arno},
title = {{Polynomial-Time Tractable Problems over the p-Adic Numbers}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {25:1--25:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.25},
URN = {urn:nbn:de:0030-drops-241325},
doi = {10.4230/LIPIcs.MFCS.2025.25},
annote = {Keywords: p-adic numbers, existential theory, linear theory, constraint satisfaction, linear program feasibility, NP-hardness, polynomial-time algorithm}
}
Document
Computational Complexity of Covering Regular Trees
Abstract
A graph covering projection, also referred to as a locally bijective homomorphism, is a mapping between the vertices and edges of two graphs that preserves incidences and is a local bijection. This concept originates in topological graph theory but has also found applications in combinatorics and theoretical computer science. In this paper we consider undirected graphs in the most general setting - graphs may contain multiple edges, loops, and semi-edges. This is in line with recent trends in topological graph theory and mathematical physics.
We advance the study of the computational complexity of the H-Cover problem, which asks whether an input graph allows a covering projection onto a parameter graph H. The quest for a complete characterization started in 1990’s. Several results for simple graphs or graphs without semi-edges have been known, the role of semi-edges in the complexity setting has started to be investigated only recently. One of the most general known NP-hardness results states that H-Cover is NP-complete for every simple connected regular graph of valency greater than two. We complement this result by considering regular graphs H arising from connected acyclic graphs by adding semi-edges. Namely, we prove that any graph obtained by adding semi-edges to the vertices of a tree making it a d-regular graph with d ≥ 3, defines an NP-complete graph covering problem. In line with the so called Strong Dichotomy Conjecture, we prove that the NP-hardness holds even for simple graphs on input.
Cite as
Jan Bok, Jiří Fiala, Nikola Jedličková, and Jan Kratochvíl. Computational Complexity of Covering Regular Trees. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 26:1-26:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{bok_et_al:LIPIcs.MFCS.2025.26,
author = {Bok, Jan and Fiala, Ji\v{r}{\'\i} and Jedli\v{c}kov\'{a}, Nikola and Kratochv{\'\i}l, Jan},
title = {{Computational Complexity of Covering Regular Trees}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {26:1--26:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.26},
URN = {urn:nbn:de:0030-drops-241338},
doi = {10.4230/LIPIcs.MFCS.2025.26},
annote = {Keywords: graph cover, covering projection, semi-edges, multigraphs, complexity, constrained homomorphisms, trees}
}
Document
Cops and Robbers for Graphs on Surfaces with Crossings
Abstract
Cops and Robbers is a game played on a graph where a set of cops attempt to capture a single robber. The game proceeds in rounds, where each round first consists of the cops' turn, followed by the robber’s turn. In the first round, the cops place themselves on a subset of vertices, after which the robber chooses a vertex to place himself. From the next round onwards, in the cops' turn, every cop can choose to either stay on the same vertex or move to an adjacent vertex, and likewise the robber in his turn. The robber is considered to be captured if, at any point in time, there is some cop on the same vertex as the robber. The cops win if they can capture the robber within a finite number of rounds; else the robber wins.
A natural question in this game concerns the cop-number of a graph - the minimum number of cops needed to capture a robber. It has long been known that graphs embeddable (without crossings) on surfaces of bounded genus have bounded cop-number. In contrast, it was shown recently that the class of 1-planar graphs - graphs that can be drawn on the plane with at most one crossing per edge - does not have bounded cop-number. This paper initiates an investigation into how the distance between crossing pairs of edges influences a graph’s cop number. In particular, we look at Distance d Cops and Robbers, a variant of the classical game, where the robber is considered to be captured if there is a cop within distance d of the robber.
Let c_d(G) denote the minimum number of cops required in the graph G to capture a robber within distance d. We look at various classes of graphs, such as 1-plane graphs, k-plane graphs (graphs where each edge is crossed at most k times), and even general graph drawings, and show that if every crossing pair of edges can be connected by a path of small length, then c_d(G) is bounded, for small values of d. For example, we show that if a graph G admits a drawing in which every pair of crossing edges is contained in a path of length at most 3, then c₄(G) ≤ 21. And if the drawing permits a stronger assumption that the endpoints of every crossing induce the complete graph K₄, then c₃(G) ≤ 9. The tools and techniques that we develop in this paper are sufficiently general, enabling us to examine graphs drawn not only on the sphere but also on orientable and non-orientable surfaces.
Cite as
Prosenjit Bose, Pat Morin, and Karthik Murali. Cops and Robbers for Graphs on Surfaces with Crossings. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 27:1-27:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{bose_et_al:LIPIcs.MFCS.2025.27,
author = {Bose, Prosenjit and Morin, Pat and Murali, Karthik},
title = {{Cops and Robbers for Graphs on Surfaces with Crossings}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {27:1--27:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.27},
URN = {urn:nbn:de:0030-drops-241349},
doi = {10.4230/LIPIcs.MFCS.2025.27},
annote = {Keywords: Cops and Robbers, Crossings, 1-Planar, Surfaces}
}
Document
Graphs with No Long Claws: An Improved Bound for the Analog of the Gyárfás' Path Argument
Abstract
For a fixed integer t ⩾ 1, a (t-)long claw, denoted S_{t,t,t}, is the unique tree with three leaves, each at distance exactly t from the vertex of degree three. Majewski et al. [ICALP 2022, ACM ToCT 2024] proved an analog of the Gyárfás' path argument for S_{t,t,t}-free graphs: given an n-vertex S_{t,t,t}-free graph, one can delete neighborhoods of 𝒪(log n) vertices so that the remainder admits an extended strip decomposition (an appropriate generalization of partition into connected components) into particles of multiplicatively smaller size. In this work, we refine the argument of Majewski et al. to its arguably final form: we show that a constant number of neighborhoods suffice.
The statement of Majewski et al. is one of the two pillars of a recent quasi-polynomial time algorithm for Maximum Weight Independent Set in S_{t,t,t}-free graphs [Gartland et al., STOC 2024]; our work immediately improves the quasi-polynomial function in the running time bound. Furthermore, our result significantly simplifies known polynomial-time algorithms for Maximum Weight Independent Set in S_{t,t,t}-free graphs with an additional sparsity assumption such as bounded degree or excluding a fixed biclique as a subgraph.
Cite as
Romain Bourneuf, Jana Masaříková, Wojciech Nadara, and Marcin Pilipczuk. Graphs with No Long Claws: An Improved Bound for the Analog of the Gyárfás' Path Argument. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 28:1-28:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{bourneuf_et_al:LIPIcs.MFCS.2025.28,
author = {Bourneuf, Romain and Masa\v{r}{\'\i}kov\'{a}, Jana and Nadara, Wojciech and Pilipczuk, Marcin},
title = {{Graphs with No Long Claws: An Improved Bound for the Analog of the Gy\'{a}rf\'{a}s' Path Argument}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {28:1--28:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.28},
URN = {urn:nbn:de:0030-drops-241350},
doi = {10.4230/LIPIcs.MFCS.2025.28},
annote = {Keywords: long-claw-free graphs, extended strip decomposition, maximum weight independent set, Gy\'{a}rf\'{a}s' path, three in a tree}
}
Document
A Universal Uniform Approximation Theorem for Neural Networks
Abstract
We show the existence of a fixed recurrent network capable of approximating any computable function with arbitrary precision, provided that an encoding of the function is given in the initial input. While uniform approximation over a compact domain is a well-known property of neural networks, we go further by proving that our network ensures effective uniform approximation - simultaneously ensuring:
- Uniform approximation in the sup-norm sense, guaranteeing precision across the compact domain {[0,1]^d};
- Uniformity in the sense of computability theory (also referred to as effectivity or universality), meaning the same network works for all computable functions. Our result is obtained constructively, using original arguments. Moreover, our construction bridges computation theory with neural network approximation, providing new insights into the fundamental connections between circuit complexity and function representation.
Furthermore, this connection extends beyond computability to complexity theory. The obtained network is efficient: if a function is computable or approximable in polynomial time in the Turing machine model, then the network requires only a polynomial number of recurrences or iterations to achieve the same level of approximation, and conversely. Moreover, the recurrent network can be assumed to be very narrow, strengthening the link our results and existing models of very deep learning, where uniform approximation properties have already been established.
Cite as
Olivier Bournez, Johanne Cohen, and Adrian Wurm. A Universal Uniform Approximation Theorem for Neural Networks. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 29:1-29:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{bournez_et_al:LIPIcs.MFCS.2025.29,
author = {Bournez, Olivier and Cohen, Johanne and Wurm, Adrian},
title = {{A Universal Uniform Approximation Theorem for Neural Networks}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {29:1--29:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.29},
URN = {urn:nbn:de:0030-drops-241365},
doi = {10.4230/LIPIcs.MFCS.2025.29},
annote = {Keywords: Models of computation, Complexity theory, Formal neural networks}
}
Document
Finding Equilibria: Simpler for Pessimists, Simplest for Optimists
Abstract
We consider equilibria in multiplayer stochastic graph games with terminal-node rewards. In such games, Nash equilibria are defined assuming that each player seeks to maximise their expected payoff, ignoring their aversion or tolerance to risk. We therefore study risk-sensitive equilibria (RSEs), where the expected payoff is replaced by a risk measure. A classical risk measure in the literature is the entropic risk measure, where each player has a real valued parameter capturing their risk-averseness. We introduce the extreme risk measure, which corresponds to extreme cases of entropic risk measure, where players are either extreme optimists or extreme pessimists. Under extreme risk measure, every player is an extremist: an extreme optimist perceives their reward as the maximum payoff that can be achieved with positive probability, while an extreme pessimist expects the minimum payoff achievable with positive probability. We argue that the extreme risk measure, especially in multi-player graph based settings, is particularly relevant as they can model several real life instances such as interactions between secure systems and potential security threats, or distributed controls for safety critical systems. We prove that RSEs defined with the extreme risk measure are guaranteed to exist when all rewards are non-negative. Furthermore, we prove that the problem of deciding whether a given game contains an RSE that generates risk measures within specified intervals is decidable and NP-complete for our extreme risk measure, and even PTIME-complete when all players are extreme optimists, while that same problem is undecidable using the entropic risk measure or even the classical expected payoff. This establishes, to our knowledge, the first decidable fragment for equilibria in simple stochastic games without restrictions on strategy types or number of players.
Cite as
Léonard Brice, Thomas A. Henzinger, and K. S. Thejaswini. Finding Equilibria: Simpler for Pessimists, Simplest for Optimists. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 30:1-30:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{brice_et_al:LIPIcs.MFCS.2025.30,
author = {Brice, L\'{e}onard and Henzinger, Thomas A. and Thejaswini, K. S.},
title = {{Finding Equilibria: Simpler for Pessimists, Simplest for Optimists}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {30:1--30:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.30},
URN = {urn:nbn:de:0030-drops-241371},
doi = {10.4230/LIPIcs.MFCS.2025.30},
annote = {Keywords: Nash equilibria, stochastic games, graph games, risk-sensitive equilibria}
}
Document
Games with ω-Automatic Preference Relations
Abstract
This paper investigates Nash equilibria (NEs) in multi-player turn-based games on graphs, where player preferences are modeled as ω-automatic relations via deterministic parity automata. Unlike much of the existing literature, which focuses on specific reward functions, our results apply to any preference relation definable by an ω-automatic relation. We analyze the computational complexity of determining the existence of an NE (possibly under some constraints), verifying whether a given strategy profile forms an NE, and checking whether a specific outcome can be realized by an NE. When a (constrained) NE exists, we show that there always exists one with finite-memory strategies. Finally, we explore fundamental properties of ω-automatic relations and their implications in the existence of equilibria.
Cite as
Véronique Bruyère, Christophe Grandmont, and Jean-François Raskin. Games with ω-Automatic Preference Relations. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 31:1-31:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{bruyere_et_al:LIPIcs.MFCS.2025.31,
author = {Bruy\`{e}re, V\'{e}ronique and Grandmont, Christophe and Raskin, Jean-Fran\c{c}ois},
title = {{Games with \omega-Automatic Preference Relations}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {31:1--31:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.31},
URN = {urn:nbn:de:0030-drops-241381},
doi = {10.4230/LIPIcs.MFCS.2025.31},
annote = {Keywords: Games played on graphs, Nash equilibrium, \omega-automatic relations, \omega-recognizable relations, constrained Nash equilibria existence problem}
}
Document
A Proof of Shur’s Conjecture on the Growth of Power-Free Languages over Large Alphabets
Abstract
We settle a conjecture of Shur on an estimation of the exponential growth rates of the languages of (n/(n-1))-free words and (n/(n-1)) ^+-free words over large alphabets of size k with a correction of order O (1/(k²)).
Cite as
Vuong Bui and Matthieu Rosenfeld. A Proof of Shur’s Conjecture on the Growth of Power-Free Languages over Large Alphabets. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 32:1-32:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{bui_et_al:LIPIcs.MFCS.2025.32,
author = {Bui, Vuong and Rosenfeld, Matthieu},
title = {{A Proof of Shur’s Conjecture on the Growth of Power-Free Languages over Large Alphabets}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {32:1--32:8},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.32},
URN = {urn:nbn:de:0030-drops-241398},
doi = {10.4230/LIPIcs.MFCS.2025.32},
annote = {Keywords: power-free languages, large alphabets, Shur’s conjecture, Dejean’s conjecture}
}
Document
Hitting and Covering Affine Families of Convex Polyhedra, with Applications to Robust Optimization
Abstract
Geometric hitting set problems, in which we seek a smallest set of points that collectively hit a given set of ranges, are ubiquitous in computational geometry. Most often, the set is discrete and is given explicitly. We propose new variants of these problems, dealing with continuous families of convex polyhedra, and show that they capture decision versions of the two-level finite adaptability problem in robust optimization. We show that these problems can be solved in strongly polynomial time when the size of the hitting/covering set and the dimension of the polyhedra and the parameter space are constant. We also show that the hitting set problem can be solved in strongly quadratic time for one-parameter families of convex polyhedra in constant dimension. This leads to new tractability results for finite adaptability that are the first ones with so-called left-hand-side uncertainty, where the underlying problem is non-linear.
Cite as
Jean Cardinal, Xavier Goaoc, and Sarah Wajsbrot. Hitting and Covering Affine Families of Convex Polyhedra, with Applications to Robust Optimization. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 33:1-33:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{cardinal_et_al:LIPIcs.MFCS.2025.33,
author = {Cardinal, Jean and Goaoc, Xavier and Wajsbrot, Sarah},
title = {{Hitting and Covering Affine Families of Convex Polyhedra, with Applications to Robust Optimization}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {33:1--33:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.33},
URN = {urn:nbn:de:0030-drops-241401},
doi = {10.4230/LIPIcs.MFCS.2025.33},
annote = {Keywords: Geometric hitting set problem, Continuous families of polyhedra, Robust optimization}
}
Document
Adaptive Query Algorithms for Relational Structures Based on Homomorphism Counts
Abstract
A query algorithm based on homomorphism counts is a procedure to decide membership for a class of finite relational structures using only homomorphism count queries. A left query algorithm can ask the number of homomorphisms from any structure to the input structure and a right query algorithm can ask the number of homomorphisms from the input structure to any other structure. We systematically compare the expressive power of different types of left or right query algorithms, including non-adaptive query algorithms, adaptive query algorithms that can ask a bounded number of queries, and adaptive query algorithms that can ask an unbounded number of queries. We also consider query algorithms where the homomorphism counting is done over the Boolean semiring 𝔹, meaning that only the existence of a homomorphism is recorded, not the precise number of them.
Cite as
Balder ten Cate, Phokion G. Kolaitis, and Arnar Á. Kristjánsson. Adaptive Query Algorithms for Relational Structures Based on Homomorphism Counts. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 34:1-34:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{tencate_et_al:LIPIcs.MFCS.2025.34,
author = {ten Cate, Balder and Kolaitis, Phokion G. and Kristj\'{a}nsson, Arnar \'{A}.},
title = {{Adaptive Query Algorithms for Relational Structures Based on Homomorphism Counts}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {34:1--34:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.34},
URN = {urn:nbn:de:0030-drops-241413},
doi = {10.4230/LIPIcs.MFCS.2025.34},
annote = {Keywords: Query algorithms, homomorphisms, homomorphism counts, directed graphs, relational structures, Datalog, constraint satisfaction}
}
Document
A Note on the Complexity of Defensive Domination
Abstract
In a graph G, a k-attack A is any set of at most k vertices and 𝓁-defense D is a set of at most 𝓁 vertices. We say that defense D counters attack A if each a ∈ A can be matched to a distinct defender d ∈ D with a equal to d or a adjacent to d in G. In the defensive domination problem, we are interested in deciding, for a graph G and positive integers k and 𝓁 given on input, if there exists an 𝓁-defense that counters every possible k-attack on G. Defensive domination is a natural resource allocation problem and can be used to model network robustness and security, disaster response strategies, and redundancy designs.
The defensive domination problem is naturally in the complexity class Σ^𝖯₂. The problem was known to be NP-hard in general, and polynomial-time algorithms were found for some restricted graph classes. In this note, we prove that the defensive domination problem is Σ^𝖯₂-complete.
We also introduce a natural variant of the defensive domination problem in which the defense is allowed to be a multiset of vertices. This variant is also Σ^𝖯₂-complete, but we show that it admits a polynomial-time algorithm in the class of interval graphs. A similar result was known for the original setting in the class of proper interval graphs.
Cite as
Steven Chaplick, Grzegorz Gutowski, and Tomasz Krawczyk. A Note on the Complexity of Defensive Domination. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 35:1-35:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{chaplick_et_al:LIPIcs.MFCS.2025.35,
author = {Chaplick, Steven and Gutowski, Grzegorz and Krawczyk, Tomasz},
title = {{A Note on the Complexity of Defensive Domination}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {35:1--35:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.35},
URN = {urn:nbn:de:0030-drops-241420},
doi = {10.4230/LIPIcs.MFCS.2025.35},
annote = {Keywords: graph domination, computational complexity}
}
Document
Counting Distinct Square Substrings in Sublinear Time
Abstract
We show that the number of distinct squares in a packed string of length n over an alphabet of size σ can be computed in 𝒪(n/log_{σ}n) time in the word-RAM model of computation. This paper is the first to introduce a sublinear time algorithm for the packed version of squares counting. The packed representation of a string of length n over an alphabet of size σ is given as a sequence of 𝒪(n/ log_{σ} n) machine words in the word-RAM model (a machine word consists of ω ≥ log₂ n bits).
Previously it was known how to count distinct squares in 𝒪(n) time [Gusfield and Stoye, JCSS 2004], even for a string over an integer alphabet, see [Crochemore et al., TCS 2014; Bannai et al., CPM 2017; Charalampopoulos et al., SPIRE 2020]. We use techniques of squares extraction from runs described by Crochemore et al. [TCS 2014]. However, the packed model requires novel approaches. In particular, we need an 𝒪(n/log_{σ}n) sized representation of all long-period runs (runs with periods that are Ω(log_{σ}n)) which guarantees sublinear time counting of potentially linearly-many implied squares. The long-period runs with a string period that is periodic itself (called layer runs) are an obstacle, since their number can be Ω(n). Fortunately, the number of all other long-period runs is 𝒪(n/log_{σ}n) and we can construct an implicit representation of all long-period runs in 𝒪(n/log_{σ}n) time by adopting the insights of Amir et al. [ESA 2019], combined with sublinear time tools provided by the PILLAR model of computations in case of packed strings. We count squares in layer runs in sublinear time by exploiting combinatorial properties of types of pyramidally-shaped groups of layer runs. As a by-product, we discover several new structural properties of runs.
Another difficulty is to compute, in sublinear time, locations of Lyndon roots of runs in packed strings, which is needed for grouping of runs that can generate equal squares. To overcome this difficulty, we introduce sparse-Lyndon roots which are based on the notion of string synchronizers proposed by Kempa and Kociumaka [STOC 2019].
Cite as
Panagiotis Charalampopoulos, Manal Mohamed, Jakub Radoszewski, Wojciech Rytter, Tomasz Waleń, and Wiktor Zuba. Counting Distinct Square Substrings in Sublinear Time. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 36:1-36:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{charalampopoulos_et_al:LIPIcs.MFCS.2025.36,
author = {Charalampopoulos, Panagiotis and Mohamed, Manal and Radoszewski, Jakub and Rytter, Wojciech and Wale\'{n}, Tomasz and Zuba, Wiktor},
title = {{Counting Distinct Square Substrings in Sublinear Time}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {36:1--36:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.36},
URN = {urn:nbn:de:0030-drops-241439},
doi = {10.4230/LIPIcs.MFCS.2025.36},
annote = {Keywords: square in a string, packed model, run (maximal repetition), Lyndon word}
}
Document
New Hardness Results for Low-Rank Matrix Completion
Abstract
The low-rank matrix completion problem asks whether a given real matrix with missing values can be completed so that the resulting matrix has low rank or is close to a low-rank matrix. The completed matrix is often required to satisfy additional structural constraints, such as positive semi-definiteness or a bounded infinity norm. The problem arises in various research fields, including machine learning, statistics, and theoretical computer science, and has broad real-world applications.
This paper presents new NP-hardness results for low-rank matrix completion problems. We show that for every sufficiently large integer d and any real number ε ∈ [2^{-O(d)},1/7], given a partial matrix A with exposed values of magnitude at most 1 that admits a positive semi-definite completion of rank d, it is NP-hard to find a positive semi-definite matrix that agrees with each given value of A up to an additive error of at most ε, even when the rank is allowed to exceed d by a multiplicative factor of O (1/(ε²⋅log(1/ε))). This strengthens a result of Hardt, Meka, Raghavendra, and Weitz (COLT, 2014), which applies to multiplicative factors smaller than 2 and to ε that decays polynomially in d. We establish similar NP-hardness results for the case where the completed matrix is constrained to have a bounded infinity norm (rather than be positive semi-definite), for which all previous hardness results rely on complexity assumptions related to the Unique Games Conjecture. Our proofs involve a novel notion of nearly orthonormal representations of graphs, the concept of line digraphs, and bounds on the rank of perturbed identity matrices.
Cite as
Dror Chawin and Ishay Haviv. New Hardness Results for Low-Rank Matrix Completion. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 37:1-37:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{chawin_et_al:LIPIcs.MFCS.2025.37,
author = {Chawin, Dror and Haviv, Ishay},
title = {{New Hardness Results for Low-Rank Matrix Completion}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {37:1--37:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.37},
URN = {urn:nbn:de:0030-drops-241448},
doi = {10.4230/LIPIcs.MFCS.2025.37},
annote = {Keywords: hardness of approximation, low-rank matrix completion, graph coloring}
}
Document
The Complexity of Separability for Semilinear Sets and Parikh Automata
Abstract
In a separability problem, we are given two sets K and L from a class 𝒞, and we want to decide whether there exists a set S from a class 𝒮 such that K ⊆ S and S ∩ L = ∅. In this case, we speak of separability of sets in 𝒞 by sets in 𝒮.
We study two types of separability problems. First, we consider separability of semilinear sets (i.e. subsets of ℕ^d for some d) by sets definable by quantifier-free monadic Presburger formulas (or equivalently, the recognizable subsets of ℕ^d). Here, a formula is monadic if each atom uses at most one variable. Second, we consider separability of languages of Parikh automata by regular languages. A Parikh automaton is a machine with access to counters that can only be incremented, and have to meet a semilinear constraint at the end of the run. Both of these separability problems are known to be decidable with elementary complexity.
Our main results are that both problems are coNP-complete. In the case of semilinear sets, coNP-completeness holds regardless of whether the input sets are specified by existential Presburger formulas, quantifier-free formulas, or semilinear representations. Our results imply that recognizable separability of rational subsets of Σ* × ℕ^d (shown decidable by Choffrut and Grigorieff) is coNP-complete as well. Another application is that regularity of deterministic Parikh automata (where the target set is specified using a quantifier-free Presburger formula) is coNP-complete as well.
Cite as
Elias Rojas Collins, Chris Köcher, and Georg Zetzsche. The Complexity of Separability for Semilinear Sets and Parikh Automata. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 38:1-38:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{collins_et_al:LIPIcs.MFCS.2025.38,
author = {Collins, Elias Rojas and K\"{o}cher, Chris and Zetzsche, Georg},
title = {{The Complexity of Separability for Semilinear Sets and Parikh Automata}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {38:1--38:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.38},
URN = {urn:nbn:de:0030-drops-241457},
doi = {10.4230/LIPIcs.MFCS.2025.38},
annote = {Keywords: Vector Addition System, Separability, Regular Language}
}
Document
Right-Linear Lattices: An Algebraic Theory of ω-Regular Languages, with Fixed Points
Abstract
Alternating parity automata (APAs) provide a robust formalism for modelling infinite behaviours and play a central role in formal verification. Despite their widespread use, the algebraic theory underlying APAs has remained largely unexplored. In recent work [Anupam Das and Abhishek De, 2024], a notation for non-deterministic finite automata (NFAs) was introduced, along with a sound and complete axiomatisation of their equational theory via right-linear algebras. In this paper, we extend that line of work to the setting of infinite words. In particular, we present a dualised syntax, yielding a notation for APAs based on right-linear lattice expressions, and provide a natural axiomatisation of their equational theory with respect to the standard language model of ω-regular languages. The design of this axiomatisation is guided by the theory of fixed point logics; in fact, the completeness factors cleanly through the completeness of the linear-time μ-calculus.
Cite as
Anupam Das and Abhishek De. Right-Linear Lattices: An Algebraic Theory of ω-Regular Languages, with Fixed Points. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 39:1-39:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{das_et_al:LIPIcs.MFCS.2025.39,
author = {Das, Anupam and De, Abhishek},
title = {{Right-Linear Lattices: An Algebraic Theory of \omega-Regular Languages, with Fixed Points}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {39:1--39:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.39},
URN = {urn:nbn:de:0030-drops-241461},
doi = {10.4230/LIPIcs.MFCS.2025.39},
annote = {Keywords: omega-languages, regular languages, fixed points, Kleene algebras, right-linear grammars}
}
Document
Symmetric Proofs in the Ideal Proof System
Abstract
We consider the Ideal Proof System (IPS) introduced by Grochow and Pitassi and pose the question of which tautologies admit symmetric proofs, and of what complexity. The symmetry requirement in proofs is inspired by recent work establishing lower bounds in other symmetric models of computation. We link the existence of symmetric IPS proofs to the expressive power of logics such as fixed-point logic with counting and Choiceless Polynomial Time, specifically regarding the graph isomorphism problem. We identify relationships and tradeoffs between the symmetry of proofs and other parameters of IPS proofs such as size, degree and linearity. We study these on a number of standard families of tautologies from proof complexity and finite model theory such as the pigeonhole principle, the subset sum problem and the Cai-Fürer-Immerman graphs, exhibiting non-trivial upper bounds on the size of symmetric IPS proofs.
Cite as
Anuj Dawar, Erich Grädel, Leon Kullmann, and Benedikt Pago. Symmetric Proofs in the Ideal Proof System. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 40:1-40:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{dawar_et_al:LIPIcs.MFCS.2025.40,
author = {Dawar, Anuj and Gr\"{a}del, Erich and Kullmann, Leon and Pago, Benedikt},
title = {{Symmetric Proofs in the Ideal Proof System}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {40:1--40:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.40},
URN = {urn:nbn:de:0030-drops-241477},
doi = {10.4230/LIPIcs.MFCS.2025.40},
annote = {Keywords: proof complexity, algebraic complexity, descriptive complexity, symmetric circuits, graph isomorphism}
}
Document
Token Sliding Reconfiguration on DAGs
Abstract
Given a graph G and two independent sets of same size, the Independent Set Reconfiguration Problem under token sliding asks whether one can, in a step by step manner, transform the first independent set into the second one. In each step we must preserve the condition of independence. Further, referring to solution vertices as tokens, we are only permitted to slide a token along an edge. Until the recent work of Ito et al. [Ito et al. MFCS 2022] this problem was only considered on undirected graphs. In this work, we study reconfiguration under token sliding focusing on DAGs.
We present a complete dichotomy of intractability in regard to the depth of the DAG, by proving that this problem is NP-complete for DAGs of depth 3 and W[1]-hard for depth 4 when parameterized by the number of tokens k, and that these bounds are tight. Further, we prove that it is fixed parameter tractable on DAGs parameterized by the combination of treewidth and k. We show that this result applies to undirected graphs, when the number of times a token can visit a vertex is restricted.
Cite as
Jona Dirks and Alexandre Vigny. Token Sliding Reconfiguration on DAGs. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 41:1-41:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{dirks_et_al:LIPIcs.MFCS.2025.41,
author = {Dirks, Jona and Vigny, Alexandre},
title = {{Token Sliding Reconfiguration on DAGs}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {41:1--41:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.41},
URN = {urn:nbn:de:0030-drops-241486},
doi = {10.4230/LIPIcs.MFCS.2025.41},
annote = {Keywords: Graph theory, FPT algorithms, Reconfiguration, Independent Sets}
}
Document
Broadcasting Under Structural Restrictions
Abstract
In the Telephone Broadcast problem we are given a graph G = (V,E) with a designated source vertex s ∈ V. Our goal is to transmit a message, which is initially known only to s, to all vertices of the graph by using a process where in each round an informed vertex may transmit the message to one of its uninformed neighbors. The optimization objective is to minimize the number of rounds.
Following up on several recent works, we investigate the structurally parameterized complexity of Telephone Broadcast. In particular, we first strengthen existing NP-hardness results by showing that the problem remains NP-complete on graphs of bounded tree-depth and also on cactus graphs which are one vertex deletion away from being path forests. Motivated by this (severe) hardness, we study several other parameterizations of the problem and obtain FPT algorithms parameterized by vertex integrity (generalizing a recent FPT algorithm parameterized by vertex cover by Fomin, Fraigniaud, and Golovach [TCS 2024]) and by distance to clique, as well as FPT approximation algorithms parameterized by clique-cover and cluster vertex deletion. Furthermore, we obtain structural results that relate the length of the optimal broadcast protocol of a graph G with its pathwidth and tree-depth. By presenting a substantial improvement over the best previously known bound for pathwidth (Aminian, Kamali, Seyed-Javadi, and Sumedha [ICALP 2025]) we exponentially improve the approximation ratio achievable in polynomial time on graphs of bounded pathwidth from 𝒪(4^pw) to 𝒪(pw).
Cite as
Yudai Egami, Tatsuya Gima, Tesshu Hanaka, Yasuaki Kobayashi, Michael Lampis, Valia Mitsou, Edouard Nemery, Yota Otachi, Manolis Vasilakis, and Daniel Vaz. Broadcasting Under Structural Restrictions. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 42:1-42:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{egami_et_al:LIPIcs.MFCS.2025.42,
author = {Egami, Yudai and Gima, Tatsuya and Hanaka, Tesshu and Kobayashi, Yasuaki and Lampis, Michael and Mitsou, Valia and Nemery, Edouard and Otachi, Yota and Vasilakis, Manolis and Vaz, Daniel},
title = {{Broadcasting Under Structural Restrictions}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {42:1--42:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.42},
URN = {urn:nbn:de:0030-drops-241492},
doi = {10.4230/LIPIcs.MFCS.2025.42},
annote = {Keywords: Parameterized Complexity, Structural Graph Parameters, Telephone Broadcast}
}
Document
The Complexity of Computing Second Solutions
Abstract
We study the complexity of computing second solutions for NP search problems, i. e., given a problem instance x and a valid solution y, we have to find another valid solution y'.
Our main result shows that for typical NP decision problems, the complexity of computing second solutions is completely determined by the choice of the type of solution (i. e., the specific function problem), but independent of the underlying decision problem. More precisely, we show that for every X ∈ NP that is 1-paddable (a weak form of paddability), different choices of the type of solution lead to different second solution problems, which altogether have the same degree structure as the entire class of NP search problems (FNP). In fact, each degree of difficulty within FNP does occur as a second solution problem for X. This proves that typical NP decision problems have no intrinsic complexity w. r. t. the search for a second solution, but only the specification of the type of solution determines this complexity. This explains the empirical observation that the difficulty of computing second solutions strongly depends on the formulation of the problem.
Moreover, we show that the complexities of a search problem and its second solution variant are independent in the following sense: For all search problems A and B representing two degrees of difficulty, there exists a search problem C such that
1) C is as difficult as A and
2) computing second solutions for C is as difficult as B.
Cite as
Fabian Egidy, Christian Glaßer, and Fynn Godau. The Complexity of Computing Second Solutions. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 43:1-43:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{egidy_et_al:LIPIcs.MFCS.2025.43,
author = {Egidy, Fabian and Gla{\ss}er, Christian and Godau, Fynn},
title = {{The Complexity of Computing Second Solutions}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {43:1--43:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.43},
URN = {urn:nbn:de:0030-drops-241505},
doi = {10.4230/LIPIcs.MFCS.2025.43},
annote = {Keywords: function problems, another solution problem, turing machines}
}
Document
An EPTAS for Minimizing the Total Weighted Completion Time of Jobs with Release Dates on Uniformly Related Machines
Abstract
Scheduling of independent jobs with release dates so as to minimize the total weighted completion time is a well-known scheduling problem. Here, we study it for the classic machine environment of uniformly related machines. An efficient polynomial time approximation scheme (an EPTAS) is a family of (1+ε)-approximation algorithms where the running time is bounded by a polynomial in the input size times a function of ε > 0. For problems that are NP-hard in the strong sense, as it is the case for the problem studied here, an EPTAS is the best possible approximation scheme. We design an EPTAS for the problem by employing known techniques and introducing a large collection of new methods.
Cite as
Leah Epstein and Asaf Levin. An EPTAS for Minimizing the Total Weighted Completion Time of Jobs with Release Dates on Uniformly Related Machines. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 44:1-44:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{epstein_et_al:LIPIcs.MFCS.2025.44,
author = {Epstein, Leah and Levin, Asaf},
title = {{An EPTAS for Minimizing the Total Weighted Completion Time of Jobs with Release Dates on Uniformly Related Machines}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {44:1--44:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.44},
URN = {urn:nbn:de:0030-drops-241515},
doi = {10.4230/LIPIcs.MFCS.2025.44},
annote = {Keywords: Scheduling algorithms, Approximation schemes, Min-sum objectives}
}
Document
Regular Model Checking for Systems with Effectively Regular Reachability Relation
Abstract
Regular model checking is a well-established technique for the verification of regular transition systems (RTS): transition systems whose initial configurations and transition relation can be effectively encoded as regular languages. In 2008, To and Libkin studied RTSs in which the reachability relation (the reflexive and transitive closure of the transition relation) is also effectively regular, and showed that the recurrent reachability problem (whether a regular set L of configurations is reached infinitely often) is polynomial in the size of RTS and the transducer for the reachability relation. We extend the work of To and Libkin by studying the decidability and complexity of verifying almost-sure reachability and recurrent reachability - that is, whether L is reachable or recurrently reachable with probability 1. We then apply our results to the more common case in which only a regular overapproximation of the reachability relation is available. In particular, we extend recent complexity results on verifying safety using regular abstraction frameworks - a technique recently introduced by Czerner, the authors, and Welzel-Mohr - to liveness and almost-sure properties.
Cite as
Javier Esparza and Valentin Krasotin. Regular Model Checking for Systems with Effectively Regular Reachability Relation. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 45:1-45:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{esparza_et_al:LIPIcs.MFCS.2025.45,
author = {Esparza, Javier and Krasotin, Valentin},
title = {{Regular Model Checking for Systems with Effectively Regular Reachability Relation}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {45:1--45:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.45},
URN = {urn:nbn:de:0030-drops-241525},
doi = {10.4230/LIPIcs.MFCS.2025.45},
annote = {Keywords: Regular model checking, abstraction, inductive invariants}
}
Document
Guarding Offices with Maximum Dispersion
Abstract
We investigate the Dispersive Art Gallery Problem with vertex guards and rectangular visibility (r-visibility) for a class of orthogonal polygons that reflect the properties of real-world floor plans: these office-like polygons consist of rectangular rooms and corridors. In the dispersive variant of the Art Gallery Problem, the objective is not to minimize the number of guards but to maximize the minimum geodesic L₁-distance between any two guards, called the dispersion distance.
Our main contributions are as follows. We prove that determining whether a vertex guard set can achieve a dispersion distance of 4 in office-like polygons is NP-complete, where vertices of the polygon are restricted to integer coordinates. Additionally, we present a simple worst-case optimal algorithm that guarantees a dispersion distance of 3 in polynomial time. Our complexity result extends to polyominoes, resolving an open question posed by Rieck and Scheffer [Christian Rieck and Christian Scheffer, 2024]. When vertex coordinates are allowed to be rational, we establish analogous results, proving that achieving a dispersion distance of 2+ε is NP-hard for any ε > 0, while the classic Art Gallery Problem remains solvable in polynomial time for this class of polygons. Furthermore, we give a straightforward polynomial-time algorithm that computes worst-case optimal solutions with a dispersion distance 2.
On the other hand, for the more restricted class of hole-free independent office-like polygons, we propose a dynamic programming approach that computes optimal solutions. Moreover, we demonstrate that the problem is practically tractable for arbitrary orthogonal polygons. To this end, we compare solvers based on SAT, CP, and MIP formulations. Notably, SAT solvers efficiently compute optimal solutions for randomly generated instances with up to 1600 vertices in under 15s.
Cite as
Sándor P. Fekete, Kai Kobbe, Dominik Krupke, Joseph S. B. Mitchell, Christian Rieck, and Christian Scheffer. Guarding Offices with Maximum Dispersion. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 46:1-46:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{fekete_et_al:LIPIcs.MFCS.2025.46,
author = {Fekete, S\'{a}ndor P. and Kobbe, Kai and Krupke, Dominik and Mitchell, Joseph S. B. and Rieck, Christian and Scheffer, Christian},
title = {{Guarding Offices with Maximum Dispersion}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {46:1--46:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.46},
URN = {urn:nbn:de:0030-drops-241530},
doi = {10.4230/LIPIcs.MFCS.2025.46},
annote = {Keywords: Dispersive Art Gallery Problem, vertex guards, office-like polygons, orthogonal polygons, polyominoes, NP-completeness, worst-case optimality, dynamic programming, SAT solver}
}
Document
Quantum Programming in Polylogarithmic Time
Abstract
Polylogarithmic time delineates a relevant notion of feasibility on several classical computational models such as Boolean circuits or parallel random access machines. As far as the quantum paradigm is concerned, this notion yields the complexity class FBQPOLYLOG of functions approximable in polylogarithmic time with a quantum random access Turing machine. We introduce a quantum programming language with first-order recursive procedures, which provides the first programming language-based characterization of FBQPOLYLOG. Each program computes a function in FBQPOLYLOG (soundness) and, conversely, each function of this complexity class is computed by a program (completeness). We also provide a compilation strategy from programs to uniform families of quantum circuits of polylogarithmic depth and polynomial size, whose set of computed functions is known as qnc, and recover the well-known separation result FBQPOLYLOG ⊊ QNC.
Cite as
Florent Ferrari, Emmanuel Hainry, Romain Péchoux, and Mário Silva. Quantum Programming in Polylogarithmic Time. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 47:1-47:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{ferrari_et_al:LIPIcs.MFCS.2025.47,
author = {Ferrari, Florent and Hainry, Emmanuel and P\'{e}choux, Romain and Silva, M\'{a}rio},
title = {{Quantum Programming in Polylogarithmic Time}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {47:1--47:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.47},
URN = {urn:nbn:de:0030-drops-241547},
doi = {10.4230/LIPIcs.MFCS.2025.47},
annote = {Keywords: Quantum programming languages, Polylogarithmic time, Quantum circuits, Implicit computational complexity}
}
Document
Generalized De Bruijn Words, Invertible Necklaces, and the Burrows-Wheeler Transform
Abstract
We define generalized de Bruijn words as those words having a Burrows-Wheeler transform that is a concatenation of permutations of the alphabet. We show that generalized de Bruijn words are in 1-to-1 correspondence with Hamiltonian cycles in the generalized de Bruijn graphs, introduced in the early '80s in the context of network design. When the size of the alphabet is a prime p, we define invertible necklaces as those whose BWT-matrix is non-singular. We show that invertible necklaces of length n correspond to normal bases of the finite field 𝔽_{pⁿ}, and that they form an Abelian group isomorphic to the Reutenauer group RG_pⁿ. Using known results in abstract algebra, we can make a bridge between generalized de Bruijn words and invertible necklaces. In particular, we highlight a correspondence between binary de Bruijn words of order d+1, binary necklaces of length 2^{d} having an odd number of 1’s, invertible BWT matrices of size 2^{d}× 2^{d}, and normal bases of the finite field 𝔽_{2^{2^{d}}}.
Cite as
Gabriele Fici and Estéban Gabory. Generalized De Bruijn Words, Invertible Necklaces, and the Burrows-Wheeler Transform. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 48:1-48:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{fici_et_al:LIPIcs.MFCS.2025.48,
author = {Fici, Gabriele and Gabory, Est\'{e}ban},
title = {{Generalized De Bruijn Words, Invertible Necklaces, and the Burrows-Wheeler Transform}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {48:1--48:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.48},
URN = {urn:nbn:de:0030-drops-241555},
doi = {10.4230/LIPIcs.MFCS.2025.48},
annote = {Keywords: Burrows-Wheeler Transform, Generalized de Bruijn Word, Generalized de Bruijn Graph, Circulant Matrix, Invertible Necklace, Sandpile Group, Reutenauer Group}
}
Document
Morphisms and BWT-Run Sensitivity
Abstract
We study how the application of morphisms affects the number r of equal-letter runs in the Burrows–Wheeler Transform (BWT). This parameter has emerged as a key repetitiveness measure in compressed indexing. We focus on the notion of BWT-run sensitivity after application of morphisms. For binary alphabets, we characterize the class of injective morphisms that preserve the number of BWT-runs up to a bounded additive increase by showing that it coincides with the known class of primitivity-preserving morphisms, which are those that map primitive words to primitive words. We further prove that deciding whether a given binary morphism has bounded BWT-run sensitivity is possible in polynomial time with respect to the total length of the images of the two letters. Additionally, we explore new structural and combinatorial properties of synchronizing and recognizable morphisms. These results establish new connections between BWT-based compressibility, code theory, and symbolic dynamics.
Cite as
Gabriele Fici, Giuseppe Romana, Marinella Sciortino, and Cristian Urbina. Morphisms and BWT-Run Sensitivity. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 49:1-49:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{fici_et_al:LIPIcs.MFCS.2025.49,
author = {Fici, Gabriele and Romana, Giuseppe and Sciortino, Marinella and Urbina, Cristian},
title = {{Morphisms and BWT-Run Sensitivity}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {49:1--49:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.49},
URN = {urn:nbn:de:0030-drops-241567},
doi = {10.4230/LIPIcs.MFCS.2025.49},
annote = {Keywords: Burrows-Wheeler transform, BWT-runs, morphism, pure code, repetitiveness}
}
Document
Lexicographic Transductions of Finite Words
Abstract
Regular transductions over finite words have linear input-to-output growth. This class of transductions enjoys many characterizations, such as transductions computable by two-way transducers as well as transductions definable in MSO (in the sense of Courcelle). Recently, regular transductions have been extended by Bojańczyk to polyregular transductions, which have polynomial growth, and are characterized by pebble transducers and MSO interpretations. Another class of interest is that of transductions defined by streaming string transducers or marble transducers, which have exponential growth and are incomparable with polyregular transductions.
In this paper, we consider MSO set interpretations (MSOSI) over finite words, that were introduced by Colcombet and Loeding. MSOSI are a natural candidate for the class of "regular transductions with exponential growth", and are rather well behaved. However, MSOSI for now lacks two desirable properties that regular and polyregular transductions have. The first property is to have an automata description. This property is closely related to a second property, that of being regularity preserving, meaning preserving regular languages under inverse image.
We first show that if MSOSI are (effectively) regularity preserving then any automatic ω-word has a decidable MSO theory, an almost 20 years old conjecture of Bárány.
Our main contribution is the introduction of a class of transductions of exponential growth, which we call lexicographic transductions. We provide three different presentations for this class: first, as the closure of simple transductions (recognizable transductions) under a single operator called maplex; second, as a syntactic fragment of MSOSI (but the regular languages are given by automata instead of formulas); and third, we give an automaton based model called nested marble transducers, which generalize both marble transducers and pebble transducers. We show that this class enjoys many nice properties including being regularity preserving.
Cite as
Emmanuel Filiot, Nathan Lhote, and Pierre-Alain Reynier. Lexicographic Transductions of Finite Words. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 50:1-50:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{filiot_et_al:LIPIcs.MFCS.2025.50,
author = {Filiot, Emmanuel and Lhote, Nathan and Reynier, Pierre-Alain},
title = {{Lexicographic Transductions of Finite Words}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {50:1--50:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.50},
URN = {urn:nbn:de:0030-drops-241572},
doi = {10.4230/LIPIcs.MFCS.2025.50},
annote = {Keywords: Transducers, Automata, MSO, Logical interpretations, Automatic structures}
}
Document
Word Structures and Their Automatic Presentations
Abstract
We study automatic presentations of the structures (ℕ; S), (ℕ; E_S), (ℕ; ≤), and their expansions by a unary predicate U. Here S is the successor function, E_S is the undirected version of S, and ≤ is the natural order. We call these structures word structures. Our goal is three-fold. First, we study the isomorphism problem for automatic word structures by focusing on the following three problems. The first problem asks to design an algorithm that, given an automatic structure A, decides if A is isomorphic to (ℕ; S). The second asks to design an algorithm that, given two automatic presentations of (ℕ; S, U₁) and (ℕ; S, U₂), where U₁ and U₂ are unary predicates, decides if these structures are isomorphic. The third problem investigates if there is an algorithm that, given two automatic presentations of (ℕ; ≤, U₁) and (ℕ; ≤, U₂), decides whether U₁ ∩ U₂ ≠ ∅. We show that these problems are undecidable. Next, we study intrinsic regularity of the function S in the structure Path_ω = (ℕ; E_S). We build an automatic presentation of Path_ω in which S is not regular. This implies that S is not intrinsically regular in Path_ω. For U ⊆ ℕ, let d_U be the function that computes the distances between the consecutive elements of U. We build automatic presentations of (ℕ; ≤, U) where d_U can realise logarithmic, radical, intermediate, and exponential functions.
Cite as
Xiaoyang Gong, Bakh Khoussainov, and Yuyang Zhuge. Word Structures and Their Automatic Presentations. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 51:1-51:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{gong_et_al:LIPIcs.MFCS.2025.51,
author = {Gong, Xiaoyang and Khoussainov, Bakh and Zhuge, Yuyang},
title = {{Word Structures and Their Automatic Presentations}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {51:1--51:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.51},
URN = {urn:nbn:de:0030-drops-241581},
doi = {10.4230/LIPIcs.MFCS.2025.51},
annote = {Keywords: Automatic structures, the isomorphism problem, decidability, undecidability, regular relations}
}
Document
On the Complexity of Recoverable Robust Optimization in the Polynomial Hierarchy
Abstract
Recoverable robust optimization is a popular multi-stage approach, in which it is possible to adjust a first-stage solution after the uncertain cost scenario is revealed. We consider recoverable robust optimization in combination with discrete budgeted uncertainty. In this setting, it seems plausible that many problems become Σ^p₃-complete and therefore it is impossible to find compact IP formulations of them (unless the unlikely conjecture NP = Σ^p₃ holds). Even though this seems plausible, few concrete results of this kind are known. In this paper, we fill that gap of knowledge. We consider recoverable robust optimization for the nominal problems of Sat, 3Sat, vertex cover, dominating set, set cover, hitting set, feedback vertex set, feedback arc set, uncapacitated facility location, p-center, p-median, independent set, clique, subset sum, knapsack, partition, scheduling, Hamiltonian path/cycle (directed/undirected), TSP, k-directed disjoint path (k ≥ 2), and Steiner tree. We show that for each of these problems, and for each of three widely used distance measures, the recoverable robust problem becomes Σ^p₃-complete. Concretely, we show that all these problems share a certain abstract property and prove that this property implies that their robust recoverable counterpart is Σ^p₃-complete. This reveals the insight that all the above problems are Σ^p₃-complete "for the same reason". Our result extends a recent framework by Grüne and Wulf.
Cite as
Christoph Grüne and Lasse Wulf. On the Complexity of Recoverable Robust Optimization in the Polynomial Hierarchy. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 52:1-52:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{grune_et_al:LIPIcs.MFCS.2025.52,
author = {Gr\"{u}ne, Christoph and Wulf, Lasse},
title = {{On the Complexity of Recoverable Robust Optimization in the Polynomial Hierarchy}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {52:1--52:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.52},
URN = {urn:nbn:de:0030-drops-241596},
doi = {10.4230/LIPIcs.MFCS.2025.52},
annote = {Keywords: Complexity, Robust Optimization, Recoverable Robust Optimization, Two-Stage Problems, Polynomial Hierarchy, Sigma 2, Sigma 3}
}
Document
Wait-Only Broadcast Protocols Are Easier to Verify
Abstract
We study networks of processes that all execute the same finite-state protocol and communicate via broadcasts. We are interested in two problems with a parameterized number of processes: the synchronization problem which asks whether there is an execution which puts all processes on a given state; and the repeated coverability problem which asks if there is an infinite execution where a given transition is taken infinitely often. Since both problems are undecidable in the general case, we investigate those problems when the protocol is Wait-Only, i.e., it has no state from which a process can both broadcast and receive messages. We establish that the synchronization problem becomes Ackermann-complete, and the repeated coverability problem is in ExpSpace and PSpace-hard.
Cite as
Lucie Guillou, Arnaud Sangnier, and Nathalie Sznajder. Wait-Only Broadcast Protocols Are Easier to Verify. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 53:1-53:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{guillou_et_al:LIPIcs.MFCS.2025.53,
author = {Guillou, Lucie and Sangnier, Arnaud and Sznajder, Nathalie},
title = {{Wait-Only Broadcast Protocols Are Easier to Verify}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {53:1--53:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.53},
URN = {urn:nbn:de:0030-drops-241609},
doi = {10.4230/LIPIcs.MFCS.2025.53},
annote = {Keywords: Parameterised verification, Reachability, Broadcast}
}
Document
Quasipolynomial-Time Deterministic Kernelization and (Gammoid) Representation
Abstract
In this paper, we suggest to extend the notion of a kernel to permit the kernelization algorithm to be executed in quasi-polynomial time rather than polynomial time. So far, we are only aware of one work that addressed this negatively, showing that some lower bounds on kernel sizes proved for kernelization also hold when quasi-polynomial time complexity is allowed. When we, anyway, deal with an NP-hard problem, sacrificing polynomial time in preprocessing for quasi-polynomial time may often not be a big deal, but, of course, the question is - does it give us more power? The only known work, mentioned above, seems to suggest that the answer is "no". In this paper, we show that this is not the case - in particular, we show that this notion is extremely powerful for derandomization. Some of the most basic kernelization algorithms in the field are based on inherently randomized tools whose derandomization is a huge problem that has remained (and may still remain) open for many decades. Still, some breakthrough advances for derandomization in quasi-polynomial time have been made. Can we harness these advancements to design quasi-polynomial deterministic kernelization algorithms for basic problems in the field? To this end, we revisit the question of deterministic polynomial-time computation of a linear representation of transversal matroids and gammoids, which is a longstanding open problem. We present a deterministic computation of a representation matrix of a transversal matroid in time quasipolynomial in the rank of the matroid, where each entry of the matrix can be represented in quasipolynomial (in the rank of the matroid) bits. As a corollary, we obtain a linear representation of a gammoid in deterministic quasipolynomial time and quasipolynomial bits in the size of the underlying ground set of the gammoid. In turn, as applications of our results, we present deterministic quasi-polynomial time kernels of polynomial size for several central problems in the field.
Cite as
Rohit Gurjar, Daniel Lokshtanov, Pranabendu Misra, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Quasipolynomial-Time Deterministic Kernelization and (Gammoid) Representation. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 54:1-54:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{gurjar_et_al:LIPIcs.MFCS.2025.54,
author = {Gurjar, Rohit and Lokshtanov, Daniel and Misra, Pranabendu and Panolan, Fahad and Saurabh, Saket and Zehavi, Meirav},
title = {{Quasipolynomial-Time Deterministic Kernelization and (Gammoid) Representation}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {54:1--54:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.54},
URN = {urn:nbn:de:0030-drops-241617},
doi = {10.4230/LIPIcs.MFCS.2025.54},
annote = {Keywords: Network Flows, Gammoids, Matchings, Transversal Matroids, Matroid Representation, Derandomization}
}
Document
Model-Theoretic Forcing in Transition Algebra
Abstract
We study Löwenheim-Skolem and Omitting Types theorems in Transition Algebra, a logical system obtained by enhancing many sorted first-order logic with features from dynamic logic. The sentences we consider include compositions, unions, and transitive closures of transition relations, which are treated similarly to actions in dynamic logics to define necessity and possibility operators. We show that Upward Löwenheim-Skolem theorem, any form of compactness, and joint Robinson consistency property fail due to the expressivity of transitive closures of transitions. In this non-compact many-sorted logical system, we develop a forcing technique method by generalizing the classical method of forcing used by Keisler to prove Omitting Types theorem. Instead of working within a single signature, we work with a directed diagram of signatures, which allows us to establish Downward Löwenheim-Skolem and Omitting Types theorems despite the fact that models interpret sorts as sets, possibly empty. Building on a complete system of proof rules for Transition Algebra, we extend it with additional proof rules to reason about constructor-based and/or finite transition algebras. We then establish the completeness of this extended system for a fragment of Transition Algebra obtained by restricting models to constructor-based and/or finite transition algebras.
Cite as
Go Hashimoto and Daniel Găină. Model-Theoretic Forcing in Transition Algebra. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 55:1-55:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{hashimoto_et_al:LIPIcs.MFCS.2025.55,
author = {Hashimoto, Go and G\u{a}in\u{a}, Daniel},
title = {{Model-Theoretic Forcing in Transition Algebra}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {55:1--55:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.55},
URN = {urn:nbn:de:0030-drops-241629},
doi = {10.4230/LIPIcs.MFCS.2025.55},
annote = {Keywords: institutional model theory, algebraic specification, transition algebra, forcing, omitting types property, L\"{o}wenheim-Skolem properties, completeness}
}
Document
Negated String Containment Is Decidable
Abstract
We provide a positive answer to a long-standing open question of the decidability of the not-contains string predicate. Not-contains is practically relevant, for instance in symbolic execution of string manipulating programs. Particularly, we show that the predicate ¬Contains(x₁ … x_n, y₁ … y_m), where x₁ … x_n and y₁ … y_m are sequences of string variables constrained by regular languages, is decidable. Decidability of a not-contains predicate combined with chain-free word equations and regular membership constraints follows.
Cite as
Vojtěch Havlena, Michal Hečko, Lukáš Holík, and Ondřej Lengál. Negated String Containment Is Decidable. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 56:1-56:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{havlena_et_al:LIPIcs.MFCS.2025.56,
author = {Havlena, Vojt\v{e}ch and He\v{c}ko, Michal and Hol{\'\i}k, Luk\'{a}\v{s} and Leng\'{a}l, Ond\v{r}ej},
title = {{Negated String Containment Is Decidable}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {56:1--56:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.56},
URN = {urn:nbn:de:0030-drops-241631},
doi = {10.4230/LIPIcs.MFCS.2025.56},
annote = {Keywords: not-contains, string constraints, word combinatorics, primitive word}
}
Document
Resolving Nondeterminism with Randomness
Abstract
We define and study classes of ω-regular automata for which the nondeterminism can be resolved by a policy that uses a combination of memory and randomness on any input word, based solely on the prefix read so far. We examine two settings for providing the input word to an automaton. In the first setting, called adversarial resolvability, the input word is constructed letter-by-letter by an adversary, dependent on the resolver’s previous decisions. In the second setting, called stochastic resolvability, the adversary pre-commits to an infinite word and reveals it letter-by-letter. In each setting, we require the existence of an almost-sure resolver, i.e., a policy that ensures that as long as the adversary provides a word in the language of the underlying nondeterministic automaton, the run constructed by the policy is accepting with probability 1.
The class of automata that are adversarially resolvable is the well-studied class of history-deterministic automata. The case of stochastically resolvable automata, on the other hand, defines a novel class. Restricting the class of resolvers in both settings to stochastic policies without memory introduces two additional new classes of automata. We show that the new automata classes offer interesting trade-offs between succinctness, expressivity, and computational complexity, providing a fine gradation between deterministic automata and nondeterministic automata.
Cite as
Thomas A. Henzinger, Aditya Prakash, and K. S. Thejaswini. Resolving Nondeterminism with Randomness. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 57:1-57:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{henzinger_et_al:LIPIcs.MFCS.2025.57,
author = {Henzinger, Thomas A. and Prakash, Aditya and Thejaswini, K. S.},
title = {{Resolving Nondeterminism with Randomness}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {57:1--57:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.57},
URN = {urn:nbn:de:0030-drops-241645},
doi = {10.4230/LIPIcs.MFCS.2025.57},
annote = {Keywords: \omega-regular languages, History determinism, Stochastic strategies}
}
Document
Random Permutations in Computational Complexity
Abstract
Classical results of Bennett and Gill (1981) show that with probability 1, 𝖯^A ≠ NP^A relative to a random oracle A, and with probability 1, 𝖯^π ≠ NP^π ∩ coNP^π relative to a random permutation π. Whether 𝖯^A = NP^A ∩ coNP^A holds relative to a random oracle A remains open. While the random oracle separation has been extended to specific individually random oracles-such as Martin-Löf random or resource-bounded random oracles-no analogous result is known for individually random permutations.
We introduce a new resource-bounded measure framework for analyzing individually random permutations. We define permutation martingales and permutation betting games that characterize measure-zero sets in the space of permutations, enabling formal definitions of polynomial-time random permutations, polynomial-time betting-game random permutations, and polynomial-space random permutations.
Our main result shows that 𝖯^π ≠ NP^π ∩ coNP^π for every polynomial-time betting-game random permutation π. This is the first separation result relative to individually random permutations, rather than an almost-everywhere separation. We also strengthen a quantum separation of Bennett, Bernstein, Brassard, and Vazirani (1997) by showing that NP^π ∩ coNP^π ̸ ⊆ BQP^π for every polynomial-space random permutation π.
We investigate the relationship between random permutations and random oracles. We prove that random oracles are polynomial-time reducible from random permutations. The converse-whether every random permutation is reducible from a random oracle-remains open. We show that if NP ∩ coNP is not a measurable subset of EXP, then 𝖯^A ≠ NP^A ∩ coNP^A holds with probability 1 relative to a random oracle A. Conversely, establishing this random oracle separation with time-bounded measure would imply BPP is a measure 0 subset of EXP.
Our framework builds a foundation for studying permutation-based complexity using resource-bounded measure, in direct analogy to classical work on random oracles. It raises natural questions about the power and limitations of random permutations, their relationship to random oracles, and whether individual randomness can yield new class separations.
Cite as
John M. Hitchcock, Adewale Sekoni, and Hadi Shafei. Random Permutations in Computational Complexity. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 58:1-58:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{hitchcock_et_al:LIPIcs.MFCS.2025.58,
author = {Hitchcock, John M. and Sekoni, Adewale and Shafei, Hadi},
title = {{Random Permutations in Computational Complexity}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {58:1--58:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.58},
URN = {urn:nbn:de:0030-drops-241652},
doi = {10.4230/LIPIcs.MFCS.2025.58},
annote = {Keywords: resource-bounded measure, martingales, betting games, random permutations, random oracles}
}
Document
Complexity of Anchored Crossing Number and Crossing Number of Almost Planar Graphs
Abstract
We deal with the problem of computing the exact crossing number of almost planar graphs and the closely related problem of computing the exact anchored crossing number of a pair of planar graphs. It was shown by [Cabello and Mohar, 2013] that both problems are NP-hard; although they required an unbounded number of high-degree vertices (in the first problem) or an unbounded number of anchors (in the second problem) to prove their result. Somehow surprisingly, only three vertices of degree greater than 3 altogether, or only three anchors per each of the two graphs, are sufficient to maintain hardness of these problems, as we prove here. The new result also improves the previous result on hardness of joint crossing number on surfaces by [Hliněný and Salazar, 2015]. Our result is best possible in the anchored case since the anchored crossing number of a pair of planar graphs with two anchors each is trivial, and close to being best possible in the almost planar case since the crossing number is polytime computable for almost planar graphs of maximum degree 3 [Riskin 1996, Cabello and Mohar 2011]. The complexity of crossing number of almost planar graphs with one or two vertices of degree greater than 3 is, interestingly, still wide open.
Cite as
Petr Hliněný. Complexity of Anchored Crossing Number and Crossing Number of Almost Planar Graphs. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 59:1-59:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{hlineny:LIPIcs.MFCS.2025.59,
author = {Hlin\v{e}n\'{y}, Petr},
title = {{Complexity of Anchored Crossing Number and Crossing Number of Almost Planar Graphs}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {59:1--59:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.59},
URN = {urn:nbn:de:0030-drops-241664},
doi = {10.4230/LIPIcs.MFCS.2025.59},
annote = {Keywords: Crossing number, Anchored drawing, Almost planar graph, NP-hardness}
}
Document
Reachability in Symmetric VASS
Abstract
We investigate the reachability problem in symmetric vector addition systems with states (vass), where transitions are invariant under a group of permutations of coordinates. One extremal case, the trivial groups, yields general vass. In another extremal case, the symmetric groups, we show that the reachability problem can be solved in PSpace, regardless of the dimension of input vass (to be contrasted with Ackermannian complexity in general vass). We also consider other groups, in particular alternating and cyclic ones. Furthermore, motivated by the open status of the reachability problem in data vass, we estimate the gain in complexity when the group arises as a combination of the trivial and symmetric groups.
Cite as
Łukasz Kamiński and Sławomir Lasota. Reachability in Symmetric VASS. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 60:1-60:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{kaminski_et_al:LIPIcs.MFCS.2025.60,
author = {Kami\'{n}ski, {\L}ukasz and Lasota, S{\l}awomir},
title = {{Reachability in Symmetric VASS}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {60:1--60:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.60},
URN = {urn:nbn:de:0030-drops-241678},
doi = {10.4230/LIPIcs.MFCS.2025.60},
annote = {Keywords: vector addition systems, Petri nets, reachability problem, symmetry, permutation group}
}
Document
Quantum Relaxations of CSP and Structure Isomorphism
Abstract
We investigate quantum relaxations of two key decision problems in computer science: the constraint satisfaction problem (CSP) and the structure isomorphism problem. CSP asks whether a homomorphism exists between two relational structures, while structure isomorphism seeks an isomorphism between them. In recent years, it has become increasingly apparent that many special cases of CSP can be reformulated in terms of the existence of perfect classical strategies in non-local games, a key topic of study in quantum information theory. These games have allowed us to study quantum advantage in relation to many important decision problems, such as the k-colouring problem, and the problem of solving binary constraint systems. Abramsky et al. (2017) have shown that all of these games can be seen as special instances of a non-local CSP game. Moreover, they show that perfect quantum strategies in this CSP game can be viewed as Kleisli morphisms of a graded monad on the category of relational structures, which they dub the quantum monad. In this way, the quantum monad provides a categorical characterisation of quantum advantage for the non-local CSP game.
In this work we solidify and expand the results of Abramsky et al., answering several of their open questions. Firstly, we compare the definition of quantum graph homomorphisms arising from this work with an earlier definition of the concept due to Mančinska and Roberson and show that there are graphs which exhibit quantum advantage under one definition but not the other. Our second contribution is to extend the results of Abramsky et al. which only hold in the tensor product framework of quantum mechanics to the commuting operator framework. Next, we study a non-local structure isomorphism game, which generalises the well-studied graph isomorphism game. We show how the construction of the quantum monad can be refined to provide categorical semantics for quantum strategies in this game. This results in a category where morphisms coincide with quantum homomorphisms and isomorphisms coincide with quantum isomorphisms.
Cite as
Amin Karamlou. Quantum Relaxations of CSP and Structure Isomorphism. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 61:1-61:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{karamlou:LIPIcs.MFCS.2025.61,
author = {Karamlou, Amin},
title = {{Quantum Relaxations of CSP and Structure Isomorphism}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {61:1--61:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.61},
URN = {urn:nbn:de:0030-drops-241686},
doi = {10.4230/LIPIcs.MFCS.2025.61},
annote = {Keywords: CSP, graph isomorphism, quantum information, non-local game, quantum graph homomorphism, monad}
}
Document
The Complexity of Reachability Problems in Strongly Connected Finite Automata
Abstract
Several reachability problems in finite automata, such as completeness of NFAs and synchronisation of total DFAs, correspond to fundamental properties of sets of nonnegative matrices. In particular, the two mentioned properties correspond to matrix mortality and ergodicity, which ask whether there exists a product of the input matrices that is equal to, respectively, the zero matrix and a matrix with a column of strictly positive entries only. The case where the input automaton is strongly connected (that is, the corresponding set of nonnegative matrices is irreducible) frequently appears in applications and often admits better properties than the general case. In this paper, we address the existence of such properties from the computational complexity point of view, and develop a versatile technique to show that several NL-complete problems remain NL-complete in the strongly connected case. In particular, we show that deciding if a binary total DFA is synchronising is NL-complete even if it is promised to be strongly connected, and that deciding completeness of a binary unambiguous NFA with very limited nondeterminism is NL-complete under the same promise.
Cite as
Stefan Kiefer and Andrew Ryzhikov. The Complexity of Reachability Problems in Strongly Connected Finite Automata. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 62:1-62:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{kiefer_et_al:LIPIcs.MFCS.2025.62,
author = {Kiefer, Stefan and Ryzhikov, Andrew},
title = {{The Complexity of Reachability Problems in Strongly Connected Finite Automata}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {62:1--62:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.62},
URN = {urn:nbn:de:0030-drops-241690},
doi = {10.4230/LIPIcs.MFCS.2025.62},
annote = {Keywords: unambiguous automata, nonnegative matrices, irreducible matrix sets, strongly connected automata, matrix monoids, mortality, completeness, synchronisation, ergodicity}
}
Document
Shortest Paths in Multimode Graphs
Abstract
In this work we study shortest path problems in multimode graphs, a generalization of the min-distance measure introduced by Abboud, Vassilevska W. and Wang in [SODA'16]. A multimode shortest path is the shortest path using one of multiple "modes" of transportation that cannot be combined. This represents real-world scenarios where different modes are not combinable, such as flights operated by different airline alliances. The problem arises naturally in machine learning in the context of learning with multiple embedding. More precisely, a k-multimode graph is a collection of k graphs on the same vertex set and the k-mode distance between two vertices is defined as the minimum among the distances computed in each individual graph.
We focus on approximating fundamental graph parameters on these graphs, specifically diameter and radius. In undirected multimode graphs we first show an elegant linear time 3-approximation algorithm for 2-mode diameter. We then extend this idea into a general subroutine that can be used as a part of any α-approximation, and use it to construct a 2 and 2.5 approximation algorithm for 2-mode diameter. For undirected radius, we introduce a general scheme that can compute a 3-approximation of the k-mode radius for any k and runs in near linear time in the case of k = O(1). In the directed case we establish an equivalence between approximating 2-mode diameter on DAGs and approximating the min-diameter, while for general graphs we develop novel techniques and provide a linear time algorithm to determine whether the diameter is finite.
We also develop many conditional fine-grained lower bounds for various multimode diameter and radius approximation problems. We are able to show that many of our algorithms are tight under popular fine-grained complexity hypotheses, including our linear time 3-approximation for 3-mode undirected diameter and radius. As part of this effort we propose the first extension to the Hitting Set Hypothesis [SODA'16], which we call the 𝓁-Hitting Set Hypothesis. We use this hypothesis to prove the first parameterized lower bound tradeoff for radius approximation algorithms.
Cite as
Yael Kirkpatrick and Virginia Vassilevska Williams. Shortest Paths in Multimode Graphs. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 63:1-63:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{kirkpatrick_et_al:LIPIcs.MFCS.2025.63,
author = {Kirkpatrick, Yael and Vassilevska Williams, Virginia},
title = {{Shortest Paths in Multimode Graphs}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {63:1--63:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.63},
URN = {urn:nbn:de:0030-drops-241703},
doi = {10.4230/LIPIcs.MFCS.2025.63},
annote = {Keywords: Graph Algorithms, Shortest Paths, Diameter, Radius, Fine-Grained Complexity}
}
Document
Positional-Player Games
Abstract
In reactive synthesis, we transform a specification to a system that satisfies the specification in all environments. For specifications in linear-temporal logic, research on bounded synthesis, where the sizes of the system and the environment are bounded, captures realistic settings and has lead to algorithms of improved complexity and implementability. In the game-based {a}pproach to synthesis, the system and its environment are modeled by strategies in a two-player game with an ω-regular objective, induced by the specification. There, bounded synthesis corresponds to bounding the memory of the strategies of the players. The memory requirement for various objectives has been extensively studied. In particular, researchers have identified positional objectives, where the winning player can follow a memoryless strategy - one that needs no memory.
In this work we study bounded synthesis in the game setting. Specifically, we define and study positional-player games, in which one or both players are restricted to memoryless strategies, which correspond to non-intrusive control in various applications. We study positional-player games with Rabin, Streett, and Muller objectives, as well as with weighted multiple Büchi and reachability objectives. Our contribution covers their theoretical properties as well as a complete picture of the complexity of deciding the game in the various settings.
Cite as
Orna Kupferman and Noam Shenwald. Positional-Player Games. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 64:1-64:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{kupferman_et_al:LIPIcs.MFCS.2025.64,
author = {Kupferman, Orna and Shenwald, Noam},
title = {{Positional-Player Games}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {64:1--64:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.64},
URN = {urn:nbn:de:0030-drops-241719},
doi = {10.4230/LIPIcs.MFCS.2025.64},
annote = {Keywords: Games, \omega-Regular Objectives, Memory, Complexity}
}
Document
Parameterized Spanning Tree Congestion
Abstract
In this paper we study the Spanning Tree Congestion problem, where we are given an undirected graph G = (V,E) and are asked to find a spanning tree T of minimum maximum congestion. Here, the congestion of an edge e ∈ T is the number of edges uv ∈ E such that the (unique) path from u to v in T traverses e. We consider this well-studied NP-hard problem from the point of view of (structural) parameterized complexity and obtain the following results:
- We resolve a natural open problem by showing that Spanning Tree Congestion is not FPT parameterized by treewidth (under standard assumptions). More strongly, we present a generic reduction which applies to (almost) any parameter of the form "vertex-deletion distance to class 𝒞", thus obtaining W[1]-hardness for more restricted parameters, including tree-depth plus feedback vertex set, or incomparable to treewidth, such as twin cover. Via a slight tweak of the same reduction we also show that the problem is NP-complete on graphs of modular-width 4.
- Even though it is known that Spanning Tree Congestion remains NP-hard on instances with only one vertex of unbounded degree, it is currently open whether the problem remains hard on bounded-degree graphs. We resolve this question by showing NP-hardness on graphs of maximum degree 8.
- Complementing the problem’s W[1]-hardness for treewidth, we formulate an algorithm that runs in time roughly {(k+w)}^{𝒪(w)}, where k is the desired congestion and w the treewidth, improving a previous argument for parameter k+w that was based on Courcelle’s theorem. This explicit algorithm pays off in two ways: it allows us to obtain an FPT approximation scheme for parameter treewidth, that is, a (1+ε)-approximation running in time roughly {(w/ε)}^{𝒪(w)}; and it leads to an exact FPT algorithm for parameter clique-width+k via a Win/Win argument.
- Finally, motivated by the problem’s hardness for most standard structural parameters, we present FPT algorithms for several more restricted cases, namely, for the parameters vertex-deletion distance to clique; vertex integrity; and feedback edge set, in the latter case also achieving a single-exponential running time dependence on the parameter.
Cite as
Michael Lampis, Valia Mitsou, Edouard Nemery, Yota Otachi, Manolis Vasilakis, and Daniel Vaz. Parameterized Spanning Tree Congestion. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 65:1-65:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{lampis_et_al:LIPIcs.MFCS.2025.65,
author = {Lampis, Michael and Mitsou, Valia and Nemery, Edouard and Otachi, Yota and Vasilakis, Manolis and Vaz, Daniel},
title = {{Parameterized Spanning Tree Congestion}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {65:1--65:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.65},
URN = {urn:nbn:de:0030-drops-241724},
doi = {10.4230/LIPIcs.MFCS.2025.65},
annote = {Keywords: Parameterized Complexity, Treewidth, Graph Width Parameters}
}
Document
Deciding Regular Games: a Playground for Exponential Time Algorithms
Abstract
Regular games form a well-established class of games for analysis and synthesis of reactive systems. They include colored Muller games, McNaughton games, Muller games, Rabin games, and Streett games. These games are played on directed graphs G where Player 0 and Player 1 play by generating an infinite path ρ through the graph. The winner is determined by specifications put on the set X of vertices in ρ that occur infinitely often. These games are determined, enabling the partitioning of G into two sets Win₀ and Win₁ of winning positions for Player 0 and Player 1, respectively. Numerous algorithms exist that decide instances of regular games, e.g., Muller games, by computing Win₀ and Win₁. In this paper we aim to find general principles for designing uniform algorithms that decide all regular games. For this we utilize various recursive and dynamic programming algorithms that leverage standard notions such as subgames and traps. Importantly, we show that our techniques improve or match the performances of existing algorithms for many instances of regular games.
Cite as
Zihui Liang, Bakh Khoussainov, and Mingyu Xiao. Deciding Regular Games: a Playground for Exponential Time Algorithms. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 66:1-66:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{liang_et_al:LIPIcs.MFCS.2025.66,
author = {Liang, Zihui and Khoussainov, Bakh and Xiao, Mingyu},
title = {{Deciding Regular Games: a Playground for Exponential Time Algorithms}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {66:1--66:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.66},
URN = {urn:nbn:de:0030-drops-241732},
doi = {10.4230/LIPIcs.MFCS.2025.66},
annote = {Keywords: Regular games, colored Muller games, Rabin games, McNaughton games, Muller games, deciding games}
}
Document
#SAT-Algorithms for Classes of Threshold Circuits Based on Probabilistic Rank
Abstract
There is a large body of work that shows how to leverage lower bound techniques for circuit classes to obtain satisfiability algorithms that run in better than brute-force time [Ramamohan Paturi et al., 1997; Ryan Williams, 2014]. For circuits with threshold gates, there are several such algorithms based on either
- Probabilistic Representations by low-degree polynomials, which allow for the use of fast polynomial evaluation algorithms, or
- Low rank, which allows for an efficient reduction to rectangular matrix multiplication. In this paper, we use a related notion of probabilistic rank to obtain satisfiability algorithms for circuit classes contained in ACC⁰∘3-PTF, i.e. constant-depth circuits with modular counting gates and a single layer of degree-3 polynomial threshold functions.
Even for the special case of a single 3-PTF, it is not clear how to use either of the above two strategies to get a non-trivial satisfiability algorithm. The best known algorithm in this case previously was based on memoization and yields worse guarantees than our algorithm.
Cite as
Nutan Limaye, Adarsh Srinivasan, and Srikanth Srinivasan. #SAT-Algorithms for Classes of Threshold Circuits Based on Probabilistic Rank. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 67:1-67:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{limaye_et_al:LIPIcs.MFCS.2025.67,
author = {Limaye, Nutan and Srinivasan, Adarsh and Srinivasan, Srikanth},
title = {{#SAT-Algorithms for Classes of Threshold Circuits Based on Probabilistic Rank}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {67:1--67:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.67},
URN = {urn:nbn:de:0030-drops-241744},
doi = {10.4230/LIPIcs.MFCS.2025.67},
annote = {Keywords: probabilistic polynomials, probabilistic rank, circuit satisfiability, circuit lower bounds, polynomial method, threshold circuits}
}
Document
Quantitative Monoidal Algebra: Axiomatising Distance with String Diagrams
Abstract
String diagrammatic calculi have become increasingly popular in fields such as quantum theory, circuit theory, probabilistic programming, and machine learning, where they enable resource-sensitive and compositional algebraic analysis. Traditionally, the equations of diagrammatic calculi only axiomatise exact semantic equality. However, reasoning in these domains often involves approximations rather than strict equivalences.
In this work, we develop a quantitative framework for diagrammatic calculi, where one may axiomatise notions of distance between string diagrams. Unlike similar approaches, such as the quantitative theories introduced by Mardare et al., this requires us to work in a monoidal rather than a cartesian setting. We define a suitable notion of monoidal theory, the syntactic category it freely generates, and its models, where the concept of distance is established via enrichment over a quantale. To illustrate the framework, we provide examples from probabilistic and linear systems analysis.
Cite as
Gabriele Lobbia, Wojciech Różowski, Ralph Sarkis, and Fabio Zanasi. Quantitative Monoidal Algebra: Axiomatising Distance with String Diagrams. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 68:1-68:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{lobbia_et_al:LIPIcs.MFCS.2025.68,
author = {Lobbia, Gabriele and R\'{o}\.{z}owski, Wojciech and Sarkis, Ralph and Zanasi, Fabio},
title = {{Quantitative Monoidal Algebra: Axiomatising Distance with String Diagrams}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {68:1--68:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.68},
URN = {urn:nbn:de:0030-drops-241759},
doi = {10.4230/LIPIcs.MFCS.2025.68},
annote = {Keywords: string diagram, symmetric monoidal category, quantitative algebraic theory, quantale, metric}
}
Document
FO-Query Enumeration over SLP-Compressed Structures of Bounded Degree
Abstract
Enumerating the result set of a first-order query over a relational structure of bounded degree can be done with linear preprocessing and constant delay. In this work, we extend this result towards the compressed perspective where the structure is given in a potentially highly compressed form by a straight-line program (SLP). Our main result is an algorithm that enumerates the result set of a first-order query over a structure of bounded degree that is represented by an SLP satisfying the so-called apex condition. For a fixed formula, the enumeration algorithm has constant delay and needs a preprocessing time that is linear in the size of the SLP.
Cite as
Markus Lohrey, Sebastian Maneth, and Markus L. Schmid. FO-Query Enumeration over SLP-Compressed Structures of Bounded Degree. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 69:1-69:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{lohrey_et_al:LIPIcs.MFCS.2025.69,
author = {Lohrey, Markus and Maneth, Sebastian and Schmid, Markus L.},
title = {{FO-Query Enumeration over SLP-Compressed Structures of Bounded Degree}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {69:1--69:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.69},
URN = {urn:nbn:de:0030-drops-241760},
doi = {10.4230/LIPIcs.MFCS.2025.69},
annote = {Keywords: Enumeration algorithms, FO-logic, query evaluation over compressed data}
}
Document
Labelled Well Quasi Ordered Classes of Bounded Linear Clique-Width
Abstract
We construct an algorithm that inputs an MSO-interpretation from finite words to graphs, and decides if there exists a k ∈ ℕ such that the class of graphs induced by the interpretation is not well-quasi-ordered by the induced subgraph relation when vertices are freely labelled using {1, …, k}. In case no such k exists, we also prove that the class of graphs is not well-quasi-ordered by the induced subgraph relation when vertices are freely labelled using any well-quasi-ordered set of labels. As a byproduct of our analysis, we prove that for classes of bounded linear clique-width, a weak version of a conjecture by Pouzet holds.
Cite as
Aliaume Lopez. Labelled Well Quasi Ordered Classes of Bounded Linear Clique-Width. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 70:1-70:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{lopez:LIPIcs.MFCS.2025.70,
author = {Lopez, Aliaume},
title = {{Labelled Well Quasi Ordered Classes of Bounded Linear Clique-Width}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {70:1--70:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.70},
URN = {urn:nbn:de:0030-drops-241773},
doi = {10.4230/LIPIcs.MFCS.2025.70},
annote = {Keywords: well-quasi-ordering, linear clique-width, MSO transduction, automata theory}
}
Document
On Large Zeros of Linear Recurrence Sequences
Abstract
The Skolem Problem asks to determine whether a given integer linear recurrence sequence (LRS) has a zero term. This problem, whose decidability has been open for many decades, arises across a wide range of topics in computer science, including loop termination, formal languages, automata theory, and probabilistic model checking, amongst many others.
In the present paper, we introduce a notion of "large" zeros of (non-degenerate) linear recurrence sequences, i.e., zeros occurring at an index larger than a sixth-fold exponential of the size of the data defining the given LRS . We establish two main results. First, we show that large zeros are very sparse: the set of positive integers that can possibly arise as large zeros of some LRS has null density. This in turn immediately yields a Universal Skolem Set of density one, answering a question left open in the literature. Second, we define an infinite set of prime numbers, termed "good", having density one amongst all prime numbers, with the following property: for any large zero of a given LRS, there is an interval around the large zero together with an upper bound on the number of good primes possibly present in that interval. The bound in question is much lower than one would expect if good primes were distributed similarly as ordinary prime numbers, as per the Cramér model in number theory. We therefore conjecture that large zeros do not exist, which would entail decidability of the Skolem Problem.
Cite as
Florian Luca, Joël Ouaknine, and James Worrell. On Large Zeros of Linear Recurrence Sequences. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 71:1-71:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{luca_et_al:LIPIcs.MFCS.2025.71,
author = {Luca, Florian and Ouaknine, Jo\"{e}l and Worrell, James},
title = {{On Large Zeros of Linear Recurrence Sequences}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {71:1--71:11},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.71},
URN = {urn:nbn:de:0030-drops-241781},
doi = {10.4230/LIPIcs.MFCS.2025.71},
annote = {Keywords: Skolem Problem, linear recurrence sequences, decidability, Cram\'{e}r conjecture}
}
Document
One-Parametric Presburger Arithmetic Has Quantifier Elimination
Abstract
We give a quantifier elimination procedure for one-parametric Presburger arithmetic, the extension of Presburger arithmetic with the function x ↦ t ⋅ x, where t is a fixed free variable ranging over the integers. This resolves an open problem proposed in [Bogart et al., Discrete Analysis, 2017]. As conjectured in [Goodrick, Arch. Math. Logic, 2018], quantifier elimination is obtained for the extended structure featuring all integer division functions x ↦ ⌊x/(f(t))⌋, one for each integer polynomial f.
Our algorithm works by iteratively eliminating blocks of existential quantifiers. The elimination of a block builds on two sub-procedures, both running in non-deterministic polynomial time. The first one is an adaptation of a recently developed and efficient quantifier elimination procedure for Presburger arithmetic, modified to handle formulae with coefficients over the ring ℤ[t] of univariate polynomials. The second is reminiscent of the so-called "base t division method" used by Bogart et al. As a result, we deduce that the satisfiability problem for the existential fragment of one-parametric Presburger arithmetic (which encompasses a broad class of non-linear integer programs) is in NP, and that the smallest solution to a satisfiable formula in this fragment is of polynomial bit size.
Cite as
Alessio Mansutti and Mikhail R. Starchak. One-Parametric Presburger Arithmetic Has Quantifier Elimination. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 72:1-72:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{mansutti_et_al:LIPIcs.MFCS.2025.72,
author = {Mansutti, Alessio and Starchak, Mikhail R.},
title = {{One-Parametric Presburger Arithmetic Has Quantifier Elimination}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {72:1--72:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.72},
URN = {urn:nbn:de:0030-drops-241794},
doi = {10.4230/LIPIcs.MFCS.2025.72},
annote = {Keywords: decision procedures, quantifier elimination, non-linear integer arithmetic}
}
Document
Counting Locally Optimal Tours in the TSP
Abstract
We show that the problem of counting 2-optimal tours in instances of the Travelling Salesperson Problem (TSP) on complete graphs is #P-complete. In addition, we show that the expected number of 2-optimal tours in random instances of the TSP on complete graphs is O(1.2098ⁿ √{n!}). Based on numerical experiments, we conjecture that the true bound is at most O(√{n!}), which is approximately the square root of the total number of tours.
Cite as
Bodo Manthey and Jesse van Rhijn. Counting Locally Optimal Tours in the TSP. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 73:1-73:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{manthey_et_al:LIPIcs.MFCS.2025.73,
author = {Manthey, Bodo and van Rhijn, Jesse},
title = {{Counting Locally Optimal Tours in the TSP}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {73:1--73:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.73},
URN = {urn:nbn:de:0030-drops-241807},
doi = {10.4230/LIPIcs.MFCS.2025.73},
annote = {Keywords: Travelling salesman problem, probabilistic analysis, local search, heuristics, 2-opt}
}
Document
Subcoloring of (Unit) Disk Graphs
Abstract
A subcoloring of a graph is a partition of its vertex set into subsets (called colors), each inducing a disjoint union of cliques. It is a natural generalization of the classical proper coloring, in which each color must instead induce an independent set. Similarly to proper coloring, we define the subchromatic number of a graph as the minimum integer k such that it admits a subcoloring with k colors, and the corresponding problem k-Subcoloring which asks whether a graph has subchromatic number at most k. In this paper, we initiate the study of the subcoloring of (unit) disk graphs. One motivation stems from the fact that disk graphs can be seen as a dense generalization of planar graphs where, intuitively, each vertex can be blown into a large clique-much like subcoloring generalizes proper coloring. Interestingly, it can be observed that every unit disk graph admits a subcoloring with at most 7 colors. We first prove that the subchromatic number can be 3-approximated in polynomial-time in unit disk graphs. We then present several hardness results for special cases of unit disk graphs which somehow prevents the use of classical approaches for improving this result. We show in particular that 2-Subcoloring remains NP-hard in triangle-free unit disk graphs, as well as in unit disk graphs representable within a strip of bounded height. We also solve an open question of Broersma, Fomin, Nešetřil, and Woeginger (2002) by proving that 3-Subcoloring remains NP-hard in co-comparability graphs (which contain unit disk graphs representable within a strip of height √3/2). Finally, we prove that every n-vertex disk graph admits a subcoloring with at most O(log³(n)) colors and present a O(log²(n))-approximation algorithm for computing the subchromatic number of such graphs. This is achieved by defining a decomposition and a special type of co-comparability disk graph, called Δ-disk graphs, which might be of independent interest.
Cite as
Malory Marin and Rémi Watrigant. Subcoloring of (Unit) Disk Graphs. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 74:1-74:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{marin_et_al:LIPIcs.MFCS.2025.74,
author = {Marin, Malory and Watrigant, R\'{e}mi},
title = {{Subcoloring of (Unit) Disk Graphs}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {74:1--74:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.74},
URN = {urn:nbn:de:0030-drops-241811},
doi = {10.4230/LIPIcs.MFCS.2025.74},
annote = {Keywords: subcoloring, algorithms, disk graphs, unit disk graphs}
}
Document
Temporal Graph Realization with Bounded Stretch
Abstract
A periodic temporal graph, in its simplest form, is a graph in which every edge appears exactly once in the first Δ time steps, and then it reappears recurrently every Δ time steps, where Δ is a given period length. This model offers a natural abstraction of transportation networks where each transportation link connects two destinations periodically. From a network design perspective, a crucial task is to assign the time-labels on the edges in a way that optimizes some criterion. In this paper we introduce a very natural optimality criterion that captures how the temporal distances of all vertex pairs are "stretched", compared to their physical distances, i.e. their distances in the underlying static (non-temporal) graph. Given a static graph G, the task is to assign to each edge one time-label between 1 and Δ such that, in the resulting periodic temporal graph with period Δ, the duration of the fastest temporal path from any vertex u to any other vertex v is at most α times the distance between u and v in G. Here, the value of α measures how much the shortest paths are allowed to be stretched once we assign the periodic time-labels.
Our results span three different directions: First, we provide a series of approximation and NP-hardness results. Second, we provide approximation and fixed-parameter algorithms. Among them, we provide a simple polynomial-time algorithm (the radius-algorithm) which always guarantees an approximation strictly smaller than Δ, and which also computes the optimum stretch in some cases. Third, we consider a parameterized local search extension of the problem where we are given the temporal labeling of the graph, but we are allowed to change the time-labels of at most k edges; for this problem we prove that it is W[2]-hard but admits an XP algorithm with respect to k.
Cite as
George B. Mertzios, Hendrik Molter, Nils Morawietz, and Paul G. Spirakis. Temporal Graph Realization with Bounded Stretch. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 75:1-75:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{mertzios_et_al:LIPIcs.MFCS.2025.75,
author = {Mertzios, George B. and Molter, Hendrik and Morawietz, Nils and Spirakis, Paul G.},
title = {{Temporal Graph Realization with Bounded Stretch}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {75:1--75:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.75},
URN = {urn:nbn:de:0030-drops-241829},
doi = {10.4230/LIPIcs.MFCS.2025.75},
annote = {Keywords: Temporal graph, periodic temporal labeling, fastest temporal path, graph realization, temporal connectivity, stretch}
}
Document
Deciding Termination of Simple Randomized Loops
Abstract
We show that universal positive almost sure termination (UPAST) is decidable for a class of simple randomized programs, i.e., it is decidable whether the expected runtime of such a program is finite for all inputs. Our class contains all programs that consist of a single loop, with a linear loop guard and a loop body composed of two linear commuting and diagonalizable updates. In each iteration of the loop, the update to be carried out is picked at random, according to a fixed probability. We show the decidability of UPAST for this class of programs, where the program’s variables and inputs may range over various sub-semirings of the real numbers. In this way, we extend a line of research initiated by Tiwari in 2004 into the realm of randomized programs.
Cite as
Éléanore Meyer and Jürgen Giesl. Deciding Termination of Simple Randomized Loops. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 76:1-76:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{meyer_et_al:LIPIcs.MFCS.2025.76,
author = {Meyer, \'{E}l\'{e}anore and Giesl, J\"{u}rgen},
title = {{Deciding Termination of Simple Randomized Loops}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {76:1--76:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.76},
URN = {urn:nbn:de:0030-drops-241833},
doi = {10.4230/LIPIcs.MFCS.2025.76},
annote = {Keywords: decision procedures, randomized programs, linear loops, positive almost sure termination}
}
Document
Minimization of Deterministic Finite Automata Modulo the Edit Distance
Abstract
We propose a novel approach to minimization of deterministic finite automata (DFA), in which the DFA is further minimized at the expense of relaxing equality of languages to merely a similarity. As the notion of similarity of languages, we consider the edit distance between languages ℒ, ℒ', i.e., the minimal number of edits necessary to transform any word from ℒ to some word from ℒ' and vice versa.
In this paper we address two problems: minimization up to a predetermined edit distance given in the input, and minimization up to a bounded edit distance, in which there has to be an upper bound on the number of edits, but it is not specified. We show the first problem to be PSpace {}-complete and that the second problem is in Σ₂^p, and both NP-hard and coNP-hard. We show that if we limit how many strongly connected components can be visited by a single run (i.e., bounded SCC-depth), the problem becomes NP-complete. We also establish maximal subclasses of DFA over which minimization up to a bounded edit distance can be performed in polynomial time.
Additionally, we provide a succinct overview of alternative metrics for assessing language similarity.
Cite as
Jakub Michaliszyn and Jan Otop. Minimization of Deterministic Finite Automata Modulo the Edit Distance. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 77:1-77:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{michaliszyn_et_al:LIPIcs.MFCS.2025.77,
author = {Michaliszyn, Jakub and Otop, Jan},
title = {{Minimization of Deterministic Finite Automata Modulo the Edit Distance}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {77:1--77:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.77},
URN = {urn:nbn:de:0030-drops-241843},
doi = {10.4230/LIPIcs.MFCS.2025.77},
annote = {Keywords: automata theory, automata minimization, edit distance}
}
Document
Strong Keys for Tensor Isomorphism Cryptography
Abstract
Sampling a non degenerate (that is, invertible) square matrix over a finite field is easy, draw a random square matrix and discard if the determinant is zero. We address the problem in higher dimensions, and sample non degenerate boundary format tensors, which generalise square matrices. Testing degeneracy is conjectured to be hard in more than two dimensions [Hillar and Lim, 2013], precluding the "draw a random tensor and discard if degenerate" recipe. The difficulty is in computing hyperdeterminants, higher dimensional analogues of determinants. Instead, we start with a structured random non degenerate tensor and scramble it by infusing more randomness while still preserving non degeneracy. We propose two kinds of scrambling. The first is multiplication in each dimension by random invertible matrices, which preserves dimension and format. Assuming pseudo randomness of this action, which also underlies tensor isomorphism based cryptography, our samples are computationally indistinguishable from uniform non degenerate tensors. The second scrambling employs tensor convolution (that generalises multiplication by matrices) and can increase dimension. Inspired by hyperdeterminant multiplicativity, we devise a recursive sampler that uses tensor convolution to reduce the problem from arbitrary to three dimensions.
Our sampling is a candidate solution for drawing public keys in tensor isomorphism based cryptography, since non degenerate tensors elude recent weak key attacks targeting public key tensors either containing geometric structures such as "triangles" [Lars Ran and Simona Samardjiska, 2024] or being deficient in tensor rank [Gilchrist et al., 2024]. To accommodate our sampling, tensor isomorphism based schemes need to be instantiated in boundary formats such as (2k+1) × (k+1) × (k+1), away from the more familiar k × k × k cubic formats. Our sampling (along with the recent tensor trapdoor one-way functions [Anand Kumar Narayanan, 2025]) makes an enticing case to transition tensor isomorphism cryptography to boundary formats tensors, which are true analogues of square matrices.
Cite as
Anand Kumar Narayanan. Strong Keys for Tensor Isomorphism Cryptography. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 78:1-78:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{narayanan:LIPIcs.MFCS.2025.78,
author = {Narayanan, Anand Kumar},
title = {{Strong Keys for Tensor Isomorphism Cryptography}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {78:1--78:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.78},
URN = {urn:nbn:de:0030-drops-241857},
doi = {10.4230/LIPIcs.MFCS.2025.78},
annote = {Keywords: tensors, finite fields, post-quantum cryptography}
}
Document
Deciding Robust Instances of an Escape Problem for Dynamical Systems in Euclidean Space
Abstract
We study the problem of deciding whether a point escapes a closed subset of ℝ^d under the iteration of a continuous map f : ℝ^d → ℝ^d in the bit-model of real computation. We give a sound partial decision method for this problem which is complete in the sense that its halting set contains the halting set of all sound partial decision methods for the problem. Equivalently, our decision method terminates on all problem instances whose answer is robust under all sufficiently small perturbations of the function. We further show that the halting set of our algorithm is dense in the set of all problem instances. While our algorithm applies to general continuous functions, we demonstrate that it also yields complete decision methods for much more rigid function families: affine linear systems and quadratic complex polynomials. In the latter case, completeness is subject to the density of hyperbolicity conjecture in complex dynamics. This in particular yields an alternative proof of Hertling’s (2004) conditional answer to a question raised by Penrose (1989) regarding the computability of the Mandelbrot set.
Cite as
Eike Neumann. Deciding Robust Instances of an Escape Problem for Dynamical Systems in Euclidean Space. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 79:1-79:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{neumann:LIPIcs.MFCS.2025.79,
author = {Neumann, Eike},
title = {{Deciding Robust Instances of an Escape Problem for Dynamical Systems in Euclidean Space}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {79:1--79:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.79},
URN = {urn:nbn:de:0030-drops-241866},
doi = {10.4230/LIPIcs.MFCS.2025.79},
annote = {Keywords: Dynamical Systems, Computability in Analysis, Non-Linear Functions}
}
Document
On Expansions of Monadic Second-Order Logic with Dynamical Predicates
Abstract
Expansions of the monadic second-order (MSO) theory of the structure ⟨ℕ;<⟩ have been a fertile and active area of research ever since the publication of the seminal papers of Büchi and Elgot & Rabin on the subject in the 1960s. In the present paper, we establish decidability of the MSO theory of ⟨ℕ;<,P⟩, where P ranges over a large class of unary "dynamical" predicates, i.e., sets of non-negative values assumed by certain integer linear recurrence sequences. One of our key technical tools is the novel concept of (effective) prodisjunctivity, which we expect may also find independent applications further afield.
Cite as
Joris Nieuwveld and Joël Ouaknine. On Expansions of Monadic Second-Order Logic with Dynamical Predicates. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 80:1-80:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{nieuwveld_et_al:LIPIcs.MFCS.2025.80,
author = {Nieuwveld, Joris and Ouaknine, Jo\"{e}l},
title = {{On Expansions of Monadic Second-Order Logic with Dynamical Predicates}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {80:1--80:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.80},
URN = {urn:nbn:de:0030-drops-241879},
doi = {10.4230/LIPIcs.MFCS.2025.80},
annote = {Keywords: Monadic second-order logic, linear recurrence sequences, decidability, Baker’s theorem}
}
Document
Efficient Matching of Some Fundamental Regular Expressions with Backreferences
Abstract
Regular expression matching is of practical importance due to its widespread use in real-world applications. In practical use, regular expressions are often used with real-world extensions. Accordingly, the matching problem of regular expressions with real-world extensions has been actively studied in recent years, yielding steady progress. However, backreference, a popular extension supported by most modern programming languages such as Java, Python, JavaScript and others in their standard libraries for string processing, is an exception to this positive trend. In fact, it is known that the matching problem of regular expressions with backreferences (rewbs) is theoretically hard and the existence of an asymptotically fast matching algorithm for arbitrary rewbs seems unlikely. Even among currently known partial solutions, the balance between efficiency and generality remains unsatisfactory. To bridge this gap, we present an efficient matching algorithm for rewbs of the form e_0 (e)_1 e_1 \1 e_2 where e_0, e, e_1, e_2 are pure regular expressions, which are fundamental and frequently used in practical applications. It runs in quadratic time with respect to the input string length, substantially improving the best-known cubic time complexity for these rewbs. Our algorithm combines ideas from both stringology and automata theory in a novel way. We leverage two techniques from automata theory, injection and summarization, to simultaneously examine matches whose backreferenced substrings are either a fixed right-maximal repeat or its extendable prefixes, which are concepts from stringology. By further utilizing a subtle property of extendable prefixes, our algorithm correctly decides the matching problem while achieving the quadratic-time complexity.
Cite as
Taisei Nogami and Tachio Terauchi. Efficient Matching of Some Fundamental Regular Expressions with Backreferences. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 81:1-81:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{nogami_et_al:LIPIcs.MFCS.2025.81,
author = {Nogami, Taisei and Terauchi, Tachio},
title = {{Efficient Matching of Some Fundamental Regular Expressions with Backreferences}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {81:1--81:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.81},
URN = {urn:nbn:de:0030-drops-241886},
doi = {10.4230/LIPIcs.MFCS.2025.81},
annote = {Keywords: Regular expressions, Backreferences, Regex matching, NFA simulation, Suffix arrays, Right-maximal repeats}
}
Document
Tight Analysis of the Primal-Dual Method for Edge-Covering Pliable Set Families
Abstract
A classic result of Williamson, Goemans, Mihail, and Vazirani [STOC 1993: 708-717] states that the problem of covering an uncrossable set family by a min-cost edge set admits approximation ratio 2, by a primal-dual algorithm with a reverse delete phase. Bansal, Cheriyan, Grout, and Ibrahimpur [ICALP 2023: 15:1–15:19] showed that this algorithm achieves approximation ratio 16 for a larger class of so called γ-pliable set families, that have much weaker uncrossing properties. The approximation ratio 16 was improved to 10 in [Z. Nutov, 2025]. Recently, Bansal [I. Bansal, 2024] obtained approximation ratio 8 for γ-pliable families and also considered an important particular case of the family of cuts of size < k of a graph H. We will improve the approximation ratio to 7 for the former case and give a simple proof of approximation ratio 6 for the latter case. Furthermore, if H is λ-edge-connected then we will show a slightly better approximation ratio 6 - 1/(β+1), where β = ⌊(k-1)/(⌈(λ+1)/2⌉)⌋. Our analysis is supplemented by examples indicating that these approximation ratios are asymptotically tight for the primal-dual algorithm.
Cite as
Zeev Nutov. Tight Analysis of the Primal-Dual Method for Edge-Covering Pliable Set Families. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 82:1-82:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{nutov:LIPIcs.MFCS.2025.82,
author = {Nutov, Zeev},
title = {{Tight Analysis of the Primal-Dual Method for Edge-Covering Pliable Set Families}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {82:1--82:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.82},
URN = {urn:nbn:de:0030-drops-241898},
doi = {10.4230/LIPIcs.MFCS.2025.82},
annote = {Keywords: primal dual method, pliable set family, approximation algorithms}
}
Document
Three Fundamental Questions in Modern Infinite-Domain Constraint Satisfaction
Abstract
The Feder-Vardi dichotomy conjecture for Constraint Satisfaction Problems (CSPs) with finite templates, confirmed independently by Bulatov and Zhuk, has an extension to certain well-behaved infinite templates due to Bodirsky and Pinsker which remains wide open. We provide answers to three fundamental questions on the scope of the Bodirsky-Pinsker conjecture.
Our first two main results provide two simplifications of this scope, one of structural, and the other one of algebraic nature. The former simplification implies that the conjecture is equivalent to its restriction to templates without algebraicity, a crucial assumption in the most powerful classification methods. The latter yields that the higher-arity invariants of any template within its scope can be assumed to be essentially injective, and any algebraic condition characterizing any complexity class within the conjecture closed under Datalog reductions must be satisfiable by injections, thus lifting the mystery of the better applicability of certain conditions over others.
Our third main result uses the first one to show that any non-trivially tractable template within the scope serves, up to a Datalog-computable modification of it, as the witness of the tractability of a non-finitely tractable finite-domain Promise Constraint Satisfaction Problem (PCSP) by the so-called sandwich method. This generalizes a recent result of Mottet and provides a strong hitherto unknown connection between the Bodirsky-Pinsker conjecture and finite-domain PCSPs.
Cite as
Michael Pinsker, Jakub Rydval, Moritz Schöbi, and Christoph Spiess. Three Fundamental Questions in Modern Infinite-Domain Constraint Satisfaction. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 83:1-83:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{pinsker_et_al:LIPIcs.MFCS.2025.83,
author = {Pinsker, Michael and Rydval, Jakub and Sch\"{o}bi, Moritz and Spiess, Christoph},
title = {{Three Fundamental Questions in Modern Infinite-Domain Constraint Satisfaction}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {83:1--83:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.83},
URN = {urn:nbn:de:0030-drops-241903},
doi = {10.4230/LIPIcs.MFCS.2025.83},
annote = {Keywords: (Promise) Constraint Satisfaction Problem, dichotomy conjecture, polymorphism, identity, algebraicity, homogeneity, \omega-categoricity, finite boundedness, Datalog}
}
Document
Algebraic Barriers to Halving Algorithmic Information Quantities in Correlated Strings
Abstract
We study the possibility of scaling down algorithmic information quantities in tuples of correlated strings. In particular, we address a question raised by Alexander Shen: whether, for any triple of strings (a, b, c), there exists a string z such that each conditional Kolmogorov complexity C(a|z), C(b|z), C(c|z) is approximately half of the corresponding unconditional Kolmogorov complexity. We provide a negative answer to this question by constructing a triple (a, b, c) for which no such string z exists. Our construction is based on combinatorial properties of incidences in finite projective planes and relies on recent bounds for point-line incidences over prime fields, obtained using tools from additive combinatorics and algebraic methods, notably results by Bourgain-Katz-Tao and Stevens-De Zeeuw. As an application, we show that this impossibility yields lower bounds on the communication complexity of secret key agreement protocols in certain settings. These results reveal algebraic obstructions to efficient information exchange and highlight a separation in information-theoretic behavior between fields with and without proper subfields.
Cite as
Andrei Romashchenko. Algebraic Barriers to Halving Algorithmic Information Quantities in Correlated Strings. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 84:1-84:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{romashchenko:LIPIcs.MFCS.2025.84,
author = {Romashchenko, Andrei},
title = {{Algebraic Barriers to Halving Algorithmic Information Quantities in Correlated Strings}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {84:1--84:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.84},
URN = {urn:nbn:de:0030-drops-241914},
doi = {10.4230/LIPIcs.MFCS.2025.84},
annote = {Keywords: Kolmogorov complexity, algorithmic information theory, communication complexity, discrete geometry}
}
Document
Cutoff Theorems for the Equivalence of Parameterized Quantum Circuits
Abstract
Many promising quantum algorithms in economics, medical science, and material science rely on circuits that are parameterized by a large number of angles. To ensure that these algorithms are efficient, these parameterized circuits must be heavily optimized. However, most quantum circuit optimizers are not verified, so this procedure is known to be error-prone. For this reason, there is growing interest in the design of equivalence checking algorithms for parameterized quantum circuits. In this paper, we define a generalized class of parameterized circuits with arbitrary rotations and show that this problem is decidable for cyclotomic gate sets. We propose a cutoff-based procedure which reduces the problem of verifying the equivalence of parameterized quantum circuits to the problem of verifying the equivalence of finitely many parameter-free quantum circuits. Because the number of parameter-free circuits grows exponentially with the number of parameters, we also propose a probabilistic variant of the algorithm for cases when the number of parameters is intractably large. We show that our techniques extend to equivalence modulo global phase, and describe an efficient angle sampling procedure for cyclotomic gate sets.
Cite as
Neil J. Ross and Scott Wesley. Cutoff Theorems for the Equivalence of Parameterized Quantum Circuits. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 85:1-85:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{ross_et_al:LIPIcs.MFCS.2025.85,
author = {Ross, Neil J. and Wesley, Scott},
title = {{Cutoff Theorems for the Equivalence of Parameterized Quantum Circuits}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {85:1--85:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.85},
URN = {urn:nbn:de:0030-drops-241921},
doi = {10.4230/LIPIcs.MFCS.2025.85},
annote = {Keywords: Quantum Circuits, Parameterized Equivalence Checking}
}
Document
Probabilistic Finite Automaton Emptiness Is Undecidable for a Fixed Automaton
Abstract
We construct a probabilistic finite automaton (PFA) with 7 states and an input alphabet of 5 symbols for which the PFA Emptiness Problem is undecidable. The only input for the decision problem is the starting distribution. For the proof, we use reductions from special instances of the Post Correspondence Problem.
We also consider some variations: The input alphabet of the PFA can be restricted to a binary alphabet at the expense of a larger number of states. If we allow a rational output value for each state instead of a yes-no acceptance decision, the number of states can even be reduced to 6.
Cite as
Günter Rote. Probabilistic Finite Automaton Emptiness Is Undecidable for a Fixed Automaton. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 86:1-86:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{rote:LIPIcs.MFCS.2025.86,
author = {Rote, G\"{u}nter},
title = {{Probabilistic Finite Automaton Emptiness Is Undecidable for a Fixed Automaton}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {86:1--86:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.86},
URN = {urn:nbn:de:0030-drops-241930},
doi = {10.4230/LIPIcs.MFCS.2025.86},
annote = {Keywords: Probabilistic finite automaton, Undecidability, Post Correspondence Problem}
}
Document
Lazy B-Trees
Abstract
Lazy search trees (Sandlund & Wild FOCS 2020, Sandlund & Zhang SODA 2022) are sorted dictionaries whose update and query performance smoothly interpolates between that of efficient priority queues and binary search trees - automatically, depending on actual use; no adjustments are necessary to the data structure to realize the cost savings. In this paper, we design lazy B-trees, a variant of lazy search trees suitable for external memory that generalizes the speedup of B-trees over binary search trees wrt. input/output operations to the same smooth interpolation regime.
A key technical difficulty to overcome is the lack of a (fully satisfactory) external variant of biased search trees, on which lazy search trees crucially rely. We give a construction for a subset of performance guarantees sufficient to realize external-memory lazy search trees, which we deem of independent interest.
As one special case, lazy B-trees can be used as an external-memory priority queue, in which case they are competitive with some tailor-made heaps; indeed, they offer faster decrease-key and insert operations than known data structures.
Cite as
Casper Moldrup Rysgaard and Sebastian Wild. Lazy B-Trees. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 87:1-87:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{rysgaard_et_al:LIPIcs.MFCS.2025.87,
author = {Rysgaard, Casper Moldrup and Wild, Sebastian},
title = {{Lazy B-Trees}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {87:1--87:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.87},
URN = {urn:nbn:de:0030-drops-241949},
doi = {10.4230/LIPIcs.MFCS.2025.87},
annote = {Keywords: B-tree, lazy search trees, lazy updates, external memory, deferred data structures, database cracking}
}
Document
Color Refinement for Relational Structures
Abstract
Color Refinement, also known as Naive Vertex Classification, is a classical method to distinguish graphs by iteratively computing a coloring of their vertices. While it is traditionally used as an imperfect way to test for isomorphism, the algorithm has permeated many other, seemingly unrelated, areas of computer science. The method is algorithmically simple, and it has a well-understood distinguishing power: it has been logically characterized by Immerman and Lander (1990) and Cai, Fürer, Immerman (1992), who showed that it distinguishes precisely those graphs that can be distinguished by a sentence of first-order logic with counting quantifiers and only two variables. A combinatorial characterization was given by Dvořák (2010), who showed that it distinguishes precisely those graphs that differ in the number of homomorphisms from some tree.
In this paper, we introduce Relational Color Refinement (RCR, for short), a generalization of the Color Refinement method from graphs to arbitrary relational structures, whose distinguishing power admits the equivalent combinatorial and logical characterizations as Color Refinement has on graphs: we show that RCR distinguishes precisely those structures that differ in the number of homomorphisms from an acyclic connected relational structure. Further, we show that RCR distinguishes precisely those structures that are distinguished by a sentence of the guarded fragment of first-order logic with counting quantifiers. Additionally, we show that for every fixed finite relational signature, RCR can be implemented to run on structures of that signature in time O(N⋅log N), where N denotes the number of tuples present in the structure.
Cite as
Benjamin Scheidt and Nicole Schweikardt. Color Refinement for Relational Structures. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 88:1-88:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{scheidt_et_al:LIPIcs.MFCS.2025.88,
author = {Scheidt, Benjamin and Schweikardt, Nicole},
title = {{Color Refinement for Relational Structures}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {88:1--88:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.88},
URN = {urn:nbn:de:0030-drops-241958},
doi = {10.4230/LIPIcs.MFCS.2025.88},
annote = {Keywords: color refinement, counting logics, homomorphism counts, homomorphism indistinguishability, guarded logics, pebble games, relational structures, alpha-acyclicity, join-trees}
}
Document
Homomorphism Indistinguishability and Game Comonads for Restricted Conjunction and Requantification
Abstract
The notion of homomorphism indistinguishability offers a combinatorial framework for characterizing equivalence relations of graphs, in particular equivalences in counting logics within finite model theory. That is, for certain graph classes, two structures agree on all homomorphism counts from the class if and only if they satisfy the same sentences in a corresponding logic. This perspective often reveals connections between the combinatorial properties of graph classes and the syntactic structure of logical fragments. In this work, we extend this perspective to logics with restricted requantification, refining the stratification of logical resources in finite-variable counting logics. Specifically, we generalize Lovász-type theorems for these logics with either restricted conjunction or bounded quantifier-rank and present new combinatorial proofs of existing results. To this end, we introduce novel path and tree decompositions that incorporate the concept of reusability and develop characterizations based on pursuit-evasion games. Leveraging this framework, we establish that classes of bounded pathwidth and treewidth with reusability constraints are homomorphism distinguishing closed. Finally, we develop a comonadic perspective on requantification by constructing new comonads that encapsulate restricted-reusability pebble games. We show a tight correspondence between their coalgebras and path/tree decompositions, yielding categorical characterizations of reusability in graph decompositions. This unifies logical, combinatorial, and categorical perspectives on the notion of reusability.
Cite as
Georg Schindling. Homomorphism Indistinguishability and Game Comonads for Restricted Conjunction and Requantification. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 89:1-89:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{schindling:LIPIcs.MFCS.2025.89,
author = {Schindling, Georg},
title = {{Homomorphism Indistinguishability and Game Comonads for Restricted Conjunction and Requantification}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {89:1--89:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.89},
URN = {urn:nbn:de:0030-drops-241962},
doi = {10.4230/LIPIcs.MFCS.2025.89},
annote = {Keywords: homomorphism indistinguishability, game comonads, finite variable counting logic, restricted conjunction, restricted requantification, tree decomposition, path decomposition}
}
Document
Elimination Distance to Dominated Clusters
Abstract
In the Dominated Cluster Deletion problem, we are given an undirected graph G and integers k and d and the question is to decide whether there exists a set of at most k vertices whose removal results in a graph in which each connected component has a dominating set of size at most d. In the Elimination Distance to Dominated Clusters problem, we are again given an undirected graph G and integers k and d and the question is to decide whether we can recursively delete vertices up to depth k such that each remaining connected component has a dominating set of size at most d. Bentert et al. [Bentert et al., MFCS 2024] recently provided an almost complete classification of the parameterized complexity of Dominated Cluster Deletion with respect to the parameters k, d, c, and Δ, where c and Δ are the degeneracy, and the maximum degree of the input graph, respectively. In particular, they provided a non-uniform algorithm with running time f(k,d)⋅ n^{𝒪(d)}. They left as an open problem whether the problem is fixed-parameter tractable with respect to the parameter k + d + c. We provide a uniform algorithm running in time f(k,d)⋅ n^{𝒪(d)} for both Dominated Cluster Deletion and Elimination Distance to Dominated Clusters. We furthermore show that both problems are FPT when parameterized by k+d+𝓁, where 𝓁 is the semi-ladder index of the input graph, a parameter that is upper bounded and may be much smaller than the degeneracy c, positively answering the open question of Bentert et al. We further complete the picture by providing an almost full classification for the parameterized complexity and kernelization complexity of Elimination Distance to Dominated Clusters. The one difficult base case that remains open is whether Treedepth (the case d = 0) is NP-hard on graphs of bounded maximum degree.
Cite as
Nicole Schirrmacher, Sebastian Siebertz, and Alexandre Vigny. Elimination Distance to Dominated Clusters. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 90:1-90:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{schirrmacher_et_al:LIPIcs.MFCS.2025.90,
author = {Schirrmacher, Nicole and Siebertz, Sebastian and Vigny, Alexandre},
title = {{Elimination Distance to Dominated Clusters}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {90:1--90:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.90},
URN = {urn:nbn:de:0030-drops-241978},
doi = {10.4230/LIPIcs.MFCS.2025.90},
annote = {Keywords: Graph theory, Fixed-parameter algorithms, Dominated cluster, Elimination distance}
}
Document
Relative Randomness and Continuous Translation Functions
Abstract
The notions of Martin-Löf randomness and of Solovay reducibility, as well as their relations to each other, are central objects of study in algorithmic randomness. When restricted to left-c.e. reals, Solovay reducibility of α to β can be characterized [Cristian S. Calude et al., 2001] as the existence of two left approximations a₀,a₁,… and b₀,b₁,… of α and β, respectively, such that the ratios (α-a_n)/(β-b_n) are bounded from above. By a celebrated result of Kučera and Slaman, among left-c.e. reals the Martin-Löf random ones are largest with respect to Solovay reducibility. The latter result was largely improved by the Limit Theorem of Barmpalias and Lewis-Pye [George Barmpalias and Andrew Lewis-Pye, 2017], which asserts that for given left-c.e. reals α and β where β is Martin-Löf random, for all left-approximations of α and β as above, the ratios (α-a_n)/(β-b_n) converge to the same limit.
Though the original definition of Solovay reducibility applies to all reals, Solovay reducibility is considered to be badly behaved on the class of all reals. Accordingly, various variants of Solovay reducibility have been proposed, including variants defined via real-valued functions by Kumabe, Miyabe, and Suzuki [Kumabe et al., 2024] and a monotone variant by Titov [Titov, 2024]. It is known that for the monotone variant, the Limit Theorem of Barmpalias and Lewis-Pye extends to all reals [Titov, 2024]. By our main result, similarly the Limit Theorem holds for all reals with respect to the reducibility cl-open introduced by Kumabe et al. [Kumabe et al., 2024] in 2024. The result is formulated in terms of translation functions of bounded variation, and asserts that every such function from a Martin-Löf random real β to a real α is left differentiable in β. In a setting of functions that are required to be defined on the whole unit interval and not just on the reals strictly smaller than β, the differentiability of computable functions of bounded variation in every Martin-Löf random real was shown by Demuth [Demuth, 1975] in 1975; similar results for other types of computable functions and randomness notions were obtained by Brattka, Miller, and Nies [Vasco Brattka et al., 2011] in 2011 and Rute [Jason Rute, 2018] in 2018.
Furthermore, we deduce from the main result an equivalent characterization of Martin-Löf randomness on the set of left-c.e. reals in terms of cl-open-reducibility of a real to itself.
Cite as
Ivan Titov. Relative Randomness and Continuous Translation Functions. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 91:1-91:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{titov:LIPIcs.MFCS.2025.91,
author = {Titov, Ivan},
title = {{Relative Randomness and Continuous Translation Functions}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {91:1--91:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.91},
URN = {urn:nbn:de:0030-drops-241987},
doi = {10.4230/LIPIcs.MFCS.2025.91},
annote = {Keywords: Solovay reducibility, relative randomness, algorithmic randomness, Martin-L\"{o}f randomness, computable analysis, left-c.e. reals}
}
Document
On Piecewise Affine Reachability with Bellman Operators
Abstract
A piecewise affine map is one of the simplest mathematical objects exhibiting complex dynamics. The reachability problem of piecewise affine maps is as follows: Given two vectors s, t ∈ ℚ^d and a piecewise affine map f: ℚ^d → ℚ^d, is there n ∈ ℕ such that fⁿ(s) = t? Koiran, Cosnard, and Garzon show that the reachability problem of piecewise affine maps is undecidable even in dimension 2.
Most of the recent progress has been focused on decision procedures for one-dimensional piecewise affine maps, where the reachability problem has been shown to be decidable for some subclasses. However, the general undecidability discouraged research into positive results in arbitrary dimension.
In this work, we investigate a rich subclass of piecewise affine maps arising as Bellman operators of Markov decision processes (MDPs). We consider the reachability problem restricted to this subclass and examine its decidability in arbitrary dimensions. We establish that the reachability problem for Bellman operators is decidable in any dimension under either of the following conditions: (i) the target vector t is not the fixed point of the operator f; or (ii) the initial and target vectors s and t are comparable with respect to the componentwise order. Furthermore, we show that the reachability problem for two-dimensional Bellman operators is decidable for arbitrary s, t ∈ ℚ^d, in contrast to the known undecidability of reachability for general piecewise affine maps.
Cite as
Anton Varonka and Kazuki Watanabe. On Piecewise Affine Reachability with Bellman Operators. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 92:1-92:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{varonka_et_al:LIPIcs.MFCS.2025.92,
author = {Varonka, Anton and Watanabe, Kazuki},
title = {{On Piecewise Affine Reachability with Bellman Operators}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {92:1--92:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.92},
URN = {urn:nbn:de:0030-drops-241998},
doi = {10.4230/LIPIcs.MFCS.2025.92},
annote = {Keywords: piecewise affine map, reachability, value iteration, Markov decision process, Bellman operator}
}
Document
Improved Approximation Algorithms for Capacitated Vehicle Routing with Fixed Capacity
Abstract
The Capacitated Vehicle Routing Problem (CVRP) is one of the most extensively studied problems in combinatorial optimization. Based on customer demand, we distinguish three variants of CVRP: unit-demand, splittable, and unsplittable. In this paper, we consider k-CVRP in general metrics and on general graphs, where k is the vehicle capacity. All three versions are APX-hard for any fixed k ≥ 3. Assume that the approximation ratio of metric TSP is 3/2. We present a (5/2 - Θ(√{1/k}))-approximation algorithm for the splittable and unit-demand cases, and a (5/2 + ln 2 - Θ(√{1/k}))-approximation algorithm for the unsplittable case. Our approximation ratio is better than the previous results when k is less than a sufficiently large value, approximately 1.7 x 10⁷.
For small values of k, we design independent and elegant algorithms with further improvements. For the splittable and unit-demand cases, we improve the approximation ratio from 1.792 to 1.500 for k = 3, and from 1.750 to 1.500 for k = 4. For the unsplittable case, we improve the approximation ratio from 1.792 to 1.500 for k = 3, from 2.051 to 1.750 for k = 4, and from 2.249 to 2.157 for k = 5. The approximation ratio for k = 3 surprisingly achieves the same value as in the splittable case. Our techniques, such as EX-ITP - an extension of the classic ITP method, have the potential to improve algorithms for other routing problems as well.
Cite as
Jingyang Zhao and Mingyu Xiao. Improved Approximation Algorithms for Capacitated Vehicle Routing with Fixed Capacity. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 93:1-93:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{zhao_et_al:LIPIcs.MFCS.2025.93,
author = {Zhao, Jingyang and Xiao, Mingyu},
title = {{Improved Approximation Algorithms for Capacitated Vehicle Routing with Fixed Capacity}},
booktitle = {50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
pages = {93:1--93:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-388-1},
ISSN = {1868-8969},
year = {2025},
volume = {345},
editor = {Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.93},
URN = {urn:nbn:de:0030-drops-242008},
doi = {10.4230/LIPIcs.MFCS.2025.93},
annote = {Keywords: Combinatorial Optimization, Capacitated Vehicle Routing, Approximation Algorithms, Graph Algorithms}
}