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**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

We consider subsequences with gap constraints, i. e., length-k subsequences p that can be embedded into a string w such that the induced gaps (i. e., the factors of w between the positions to which p is mapped to) satisfy given gap constraints gc = (C_1, C_2, …, C_{k-1}); we call p a gc-subsequence of w. In the case where the gap constraints gc are defined by lower and upper length bounds C_i = (L^-_i, L^+_i) ∈ ℕ² and/or regular languages C_i ∈ REG, we prove tight (conditional on the orthogonal vectors (OV) hypothesis) complexity bounds for checking whether a given p is a gc-subsequence of a string w. We also consider the whole set of all gc-subsequences of a string, and investigate the complexity of the universality, equivalence and containment problems for these sets of gc-subsequences.

Joel D. Day, Maria Kosche, Florin Manea, and Markus L. Schmid. Subsequences with Gap Constraints: Complexity Bounds for Matching and Analysis Problems. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 64:1-64:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{day_et_al:LIPIcs.ISAAC.2022.64, author = {Day, Joel D. and Kosche, Maria and Manea, Florin and Schmid, Markus L.}, title = {{Subsequences with Gap Constraints: Complexity Bounds for Matching and Analysis Problems}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {64:1--64:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.64}, URN = {urn:nbn:de:0030-drops-173493}, doi = {10.4230/LIPIcs.ISAAC.2022.64}, annote = {Keywords: String algorithms, subsequences with gap constraints, pattern matching, fine-grained complexity, conditional lower bounds, parameterised complexity} }

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**Published in:** LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

A word u is a subsequence of another word w if u can be obtained from w by deleting some of its letters. In the early 1970s, Imre Simon defined the relation ∼_k (called now Simon-Congruence) as follows: two words having exactly the same set of subsequences of length at most k are ∼_k-congruent. This relation was central in defining and analysing piecewise testable languages, but has found many applications in areas such as algorithmic learning theory, databases theory, or computational linguistics. Recently, it was shown that testing whether two words are ∼_k-congruent can be done in optimal linear time. Thus, it is a natural next step to ask, for two words w and u which are not ∼_k-equivalent, what is the minimal number of edit operations that we need to perform on w in order to obtain a word which is ∼_k-equivalent to u.
In this paper, we consider this problem in a setting which seems interesting: when u is a k-subsequence universal word. A word u with alph(u) = Σ is called k-subsequence universal if the set of subsequences of length k of u contains all possible words of length k over Σ. As such, our results are a series of efficient algorithms computing the edit distance from w to the language of k-subsequence universal words.

Joel D. Day, Pamela Fleischmann, Maria Kosche, Tore Koß, Florin Manea, and Stefan Siemer. The Edit Distance to k-Subsequence Universality. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 25:1-25:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{day_et_al:LIPIcs.STACS.2021.25, author = {Day, Joel D. and Fleischmann, Pamela and Kosche, Maria and Ko{\ss}, Tore and Manea, Florin and Siemer, Stefan}, title = {{The Edit Distance to k-Subsequence Universality}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {25:1--25:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-180-1}, ISSN = {1868-8969}, year = {2021}, volume = {187}, editor = {Bl\"{a}ser, Markus and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.25}, URN = {urn:nbn:de:0030-drops-136705}, doi = {10.4230/LIPIcs.STACS.2021.25}, annote = {Keywords: Subsequence, Scattered factor, Subword, Universality, k-subsequence universality, Edit distance, Efficient algorithms} }

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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

For quadratic word equations, there exists an algorithm based on rewriting rules which generates a directed graph describing all solutions to the equation. For regular word equations - those for which each variable occurs at most once on each side of the equation - we investigate the properties of this graph, such as bounds on its diameter, size, and DAG-width, as well as providing some insights into symmetries in its structure. As a consequence, we obtain a combinatorial proof that the problem of deciding whether a regular word equation has a solution is in NP.

Joel D. Day and Florin Manea. On the Structure of Solution Sets to Regular Word Equations. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 124:1-124:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{day_et_al:LIPIcs.ICALP.2020.124, author = {Day, Joel D. and Manea, Florin}, title = {{On the Structure of Solution Sets to Regular Word Equations}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {124:1--124:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.124}, URN = {urn:nbn:de:0030-drops-125314}, doi = {10.4230/LIPIcs.ICALP.2020.124}, annote = {Keywords: Quadratic Word Equations, Regular Word Equations, String Solving, NP} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

It is a long standing conjecture that the problem of deciding whether a quadratic word equation has a solution is in NP. It has also been conjectured that the length of a minimal solution to a quadratic equation is at most exponential in the length of the equation, with the latter conjecture implying the former. We show that both conjectures hold for some natural subclasses of quadratic equations, namely the classes of regular-reversed, k-ordered, and variable-sparse quadratic equations. We also discuss a connection of our techniques to the topic of unavoidable patterns, and the possibility of exploiting this connection to produce further similar results.

Joel D. Day, Florin Manea, and Dirk Nowotka. Upper Bounds on the Length of Minimal Solutions to Certain Quadratic Word Equations. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 44:1-44:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{day_et_al:LIPIcs.MFCS.2019.44, author = {Day, Joel D. and Manea, Florin and Nowotka, Dirk}, title = {{Upper Bounds on the Length of Minimal Solutions to Certain Quadratic Word Equations}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {44:1--44:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.44}, URN = {urn:nbn:de:0030-drops-109889}, doi = {10.4230/LIPIcs.MFCS.2019.44}, annote = {Keywords: Quadratic Word Equations, Length Upper Bounds, NP, Unavoidable Patterns} }

Document

Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We investigate the locality number, a recently introduced structural parameter for strings (with applications in pattern matching with variables), and its connection to two important graph-parameters, cutwidth and pathwidth. These connections allow us to show that computing the locality number is NP-hard but fixed-parameter tractable (when the locality number or the alphabet size is treated as a parameter), and can be approximated with ratio O(sqrt{log{opt}} log n). As a by-product, we also relate cutwidth via the locality number to pathwidth, which is of independent interest, since it improves the best currently known approximation algorithm for cutwidth. In addition to these main results, we also consider the possibility of greedy-based approximation algorithms for the locality number.

Katrin Casel, Joel D. Day, Pamela Fleischmann, Tomasz Kociumaka, Florin Manea, and Markus L. Schmid. Graph and String Parameters: Connections Between Pathwidth, Cutwidth and the Locality Number (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 109:1-109:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{casel_et_al:LIPIcs.ICALP.2019.109, author = {Casel, Katrin and Day, Joel D. and Fleischmann, Pamela and Kociumaka, Tomasz and Manea, Florin and Schmid, Markus L.}, title = {{Graph and String Parameters: Connections Between Pathwidth, Cutwidth and the Locality Number}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {109:1--109:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.109}, URN = {urn:nbn:de:0030-drops-106858}, doi = {10.4230/LIPIcs.ICALP.2019.109}, annote = {Keywords: Graph and String Parameters, NP-Completeness, Approximation Algorithms} }

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**Published in:** LIPIcs, Volume 93, 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)

A pattern is a word consisting of constants from an alphabet Sigma of terminal symbols and variables from a set X. Given a pattern alpha, the decision-problem whether a given word w may be obtained by substituting the variables in \alpha for words over Sigma is called the matching problem. While this problem is, in general, NP-complete, several classes of patterns for which it can be efficiently solved are already known. We present two new classes of patterns, called k-local, and strongly-nested, and show that the respective matching problems, as well as membership can be solved efficiently for any fixed k.

Joel D. Day, Pamela Fleischmann, Florin Manea, and Dirk Nowotka. Local Patterns. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 93, pp. 24:1-24:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{day_et_al:LIPIcs.FSTTCS.2017.24, author = {Day, Joel D. and Fleischmann, Pamela and Manea, Florin and Nowotka, Dirk}, title = {{Local Patterns}}, booktitle = {37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)}, pages = {24:1--24:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-055-2}, ISSN = {1868-8969}, year = {2018}, volume = {93}, editor = {Lokam, Satya and Ramanujam, R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2017.24}, URN = {urn:nbn:de:0030-drops-84013}, doi = {10.4230/LIPIcs.FSTTCS.2017.24}, annote = {Keywords: Patterns with Variables, Local Patterns, Combinatorial Pattern Matching, Descriptive Patterns} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

We investigate the class of regular-ordered word equations. In such equations, each variable occurs at most once in each side and the order of the variables occurring in both left and right hand sides is preserved (the variables can be, however, separated by potentially distinct constant factors). Surprisingly, we obtain that solving such simple equations, even when the sides contain exactly the same variables, is NP-hard. By considerations regarding the combinatorial structure of the minimal solutions of the more general quadratic equations we obtain that the satisfiability problem for regular-ordered equations is in NP. The complexity of solving such word equations under regular constraints is also settled. Finally, we show that a related class of simple word equations, that generalises one-variable equations, is in P.

Joel D. Day, Florin Manea, and Dirk Nowotka. The Hardness of Solving Simple Word Equations. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{day_et_al:LIPIcs.MFCS.2017.18, author = {Day, Joel D. and Manea, Florin and Nowotka, Dirk}, title = {{The Hardness of Solving Simple Word Equations}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {18:1--18:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.18}, URN = {urn:nbn:de:0030-drops-81233}, doi = {10.4230/LIPIcs.MFCS.2017.18}, annote = {Keywords: Word Equations, Regular Patterns, Regular Constraints} }

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