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**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

The computational complexity of pairwise energy minimisation of N points in real space is a long-standing open problem. The idea of the potential intractability of the problem was supported by a lack of progress in finding efficient algorithms, even when restricted the integer grid approximation. In this paper we provide a firm answer to the problem on ℤ^d by showing that for a large class of pairwise energy functions the problem of periodic energy minimisation is NP-hard if the size of the period (known as a unit cell) is fixed, and is undecidable otherwise. We do so by introducing an abstraction of pairwise average energy minimisation as a mathematical problem, which covers many existing models. The most influential aspects of this work are showing for the first time: 1) undecidability of average pairwise energy minimisation in general 2) computational hardness for the most natural model with periodic boundary conditions, and 3) novel reductions for a large class of generic pairwise energy functions covering many physical abstractions at once. In particular, we develop a new tool of overlapping digital rhombuses to incorporate the properties of the physical force fields, and we connect it with classical tiling problems. Moreover, we illustrate the power of such reductions by incorporating more physical properties such as charge neutrality, and we show an inapproximability result for the extreme case of the 1D average energy minimisation problem.

Duncan Adamson, Argyrios Deligkas, Vladimir V. Gusev, and Igor Potapov. The Complexity of Periodic Energy Minimisation. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 8:1-8:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{adamson_et_al:LIPIcs.MFCS.2022.8, author = {Adamson, Duncan and Deligkas, Argyrios and Gusev, Vladimir V. and Potapov, Igor}, title = {{The Complexity of Periodic Energy Minimisation}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {8:1--8:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.8}, URN = {urn:nbn:de:0030-drops-168065}, doi = {10.4230/LIPIcs.MFCS.2022.8}, annote = {Keywords: Optimisation of periodic structures, tiling, undecidability, NP-hardness} }

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**Published in:** LIPIcs, Volume 191, 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021)

The main result of the paper is the first polynomial-time algorithm for ranking bracelets. The time-complexity of the algorithm is O(k²⋅ n⁴), where k is the size of the alphabet and n is the length of the considered bracelets. The key part of the algorithm is to compute the rank of any word with respect to the set of bracelets by finding three other ranks: the rank over all necklaces, the rank over palindromic necklaces, and the rank over enclosing apalindromic necklaces. The last two concepts are introduced in this paper. These ranks are key components to our algorithm in order to decompose the problem into parts. Additionally, this ranking procedure is used to build a polynomial-time unranking algorithm.

Duncan Adamson, Vladimir V. Gusev, Igor Potapov, and Argyrios Deligkas. Ranking Bracelets in Polynomial Time. In 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 191, pp. 4:1-4:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{adamson_et_al:LIPIcs.CPM.2021.4, author = {Adamson, Duncan and Gusev, Vladimir V. and Potapov, Igor and Deligkas, Argyrios}, title = {{Ranking Bracelets in Polynomial Time}}, booktitle = {32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021)}, pages = {4:1--4:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-186-3}, ISSN = {1868-8969}, year = {2021}, volume = {191}, editor = {Gawrychowski, Pawe{\l} and Starikovskaya, Tatiana}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2021.4}, URN = {urn:nbn:de:0030-drops-139554}, doi = {10.4230/LIPIcs.CPM.2021.4}, annote = {Keywords: Bracelets, Ranking, Necklaces} }

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

We consider the following variant of the Mortality Problem: given k x k matrices A_1, A_2, ...,A_{t}, does there exist nonnegative integers m_1, m_2, ...,m_t such that the product A_1^{m_1} A_2^{m_2} * ... * A_{t}^{m_{t}} is equal to the zero matrix? It is known that this problem is decidable when t <= 2 for matrices over algebraic numbers but becomes undecidable for sufficiently large t and k even for integral matrices.
In this paper, we prove the first decidability results for t>2. We show as one of our central results that for t=3 this problem in any dimension is Turing equivalent to the well-known Skolem problem for linear recurrence sequences. Our proof relies on the Primary Decomposition Theorem for matrices that was not used to show decidability results in matrix semigroups before. As a corollary we obtain that the above problem is decidable for t=3 and k <= 3 for matrices over algebraic numbers and for t=3 and k=4 for matrices over real algebraic numbers. Another consequence is that the set of triples (m_1,m_2,m_3) for which the equation A_1^{m_1} A_2^{m_2} A_3^{m_3} equals the zero matrix is equal to a finite union of direct products of semilinear sets.
For t=4 we show that the solution set can be non-semilinear, and thus it seems unlikely that there is a direct connection to the Skolem problem. However we prove that the problem is still decidable for upper-triangular 2 x 2 rational matrices by employing powerful tools from transcendence theory such as Baker’s theorem and S-unit equations.

Paul C. Bell, Igor Potapov, and Pavel Semukhin. On the Mortality Problem: From Multiplicative Matrix Equations to Linear Recurrence Sequences and Beyond. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 83:1-83:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bell_et_al:LIPIcs.MFCS.2019.83, author = {Bell, Paul C. and Potapov, Igor and Semukhin, Pavel}, title = {{On the Mortality Problem: From Multiplicative Matrix Equations to Linear Recurrence Sequences and Beyond}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {83:1--83:15}, 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.2019.83}, URN = {urn:nbn:de:0030-drops-110279}, doi = {10.4230/LIPIcs.MFCS.2019.83}, annote = {Keywords: Linear recurrence sequences, Skolem’s problem, mortality problem, matrix equations, primary decomposition theorem, Baker’s theorem} }

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Complete Volume

**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

LIPIcs, Volume 117, MFCS'18, Complete Volume

Igor Potapov, Paul Spirakis, and James Worrell. LIPIcs, Volume 117, MFCS'18, Complete Volume. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@Proceedings{potapov_et_al:LIPIcs.MFCS.2018, title = {{LIPIcs, Volume 117, MFCS'18, Complete Volume}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018}, URN = {urn:nbn:de:0030-drops-97459}, doi = {10.4230/LIPIcs.MFCS.2018}, annote = {Keywords: Theory of computation} }

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Front Matter

**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Front Matter, Table of Contents, Preface, Conference Organization

Igor Potapov, Paul Spirakis, and James Worrell. Front Matter, Table of Contents, Preface, Conference Organization. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 0:i-0:xx, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{potapov_et_al:LIPIcs.MFCS.2018.0, author = {Potapov, Igor and Spirakis, Paul and Worrell, James}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {0:i--0:xx}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.0}, URN = {urn:nbn:de:0030-drops-95824}, doi = {10.4230/LIPIcs.MFCS.2018.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

Document

**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We study the identity problem for matrices, i.e., whether the identity matrix is in a semigroup generated by a given set of generators. In particular we consider the identity problem for the special linear group following recent NP-completeness result for SL(2,Z) and the undecidability for SL(4,Z) generated by 48 matrices. First we show that there is no embedding from pairs of words into 3 x3 integer matrices with determinant one, i.e., into SL{(3,Z)} extending previously known result that there is no embedding into C^{2 x 2}. Apart from theoretical importance of the result it can be seen as a strong evidence that the computational problems in SL{(3,Z)} are decidable. The result excludes the most natural possibility of encoding the Post correspondence problem into SL{(3,Z)}, where the matrix products extended by the right multiplication correspond to the Turing machine simulation. Then we show that the identity problem is decidable in polynomial time for an important subgroup of SL(3,Z), the Heisenberg group H(3,Z). Furthermore, we extend the decidability result for H(n,Q) in any dimension n. Finally we are tightening the gap on decidability question for this long standing open problem by improving the undecidability result for the identity problem in SL{(4,Z)} substantially reducing the bound on the size of the generator set from 48 to 8 by developing a novel reduction technique.

Sang-Ki Ko, Reino Niskanen, and Igor Potapov. On the Identity Problem for the Special Linear Group and the Heisenberg Group. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 132:1-132:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ko_et_al:LIPIcs.ICALP.2018.132, author = {Ko, Sang-Ki and Niskanen, Reino and Potapov, Igor}, title = {{On the Identity Problem for the Special Linear Group and the Heisenberg Group}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {132:1--132:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.132}, URN = {urn:nbn:de:0030-drops-91367}, doi = {10.4230/LIPIcs.ICALP.2018.132}, annote = {Keywords: matrix semigroup, identity problem, special linear group, Heisenberg group, decidability} }

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

We consider the membership problem for matrix semigroups, which is the problem to decide whether a matrix belongs to a given finitely generated matrix semigroup.
In general, the decidability and complexity of this problem for two-dimensional matrix semigroups remains open. Recently there was a significant progress with this open problem by showing that the membership is decidable for 2x2 nonsingular integer matrices. In this paper we focus on the membership for singular integer matrices and prove that this problem is decidable for 2x2 integer matrices whose determinants are equal to 0, 1, -1 (i.e. for matrices from GL(2,Z) and any singular matrices). Our algorithm relies on a translation of numerical problems on matrices into combinatorial problems on words and conversion of the membership problem into decision problem on regular languages.

Igor Potapov and Pavel Semukhin. Membership Problem in GL(2, Z) Extended by Singular Matrices. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 44:1-44:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{potapov_et_al:LIPIcs.MFCS.2017.44, author = {Potapov, Igor and Semukhin, Pavel}, title = {{Membership Problem in GL(2, Z) Extended by Singular Matrices}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {44:1--44:13}, 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.2017.44}, URN = {urn:nbn:de:0030-drops-80737}, doi = {10.4230/LIPIcs.MFCS.2017.44}, annote = {Keywords: Matrix Semigroups, Membership Problem, General Linear Group, Singular Matrices, Automata and Formal Languages} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Robot game is a two-player vector addition game played on the integer lattice Z^n. Both players have sets of vectors and in each turn the vector chosen by a player is added to the current configuration vector of the game. One of the players, called Eve, tries to play the game from the initial configuration to the origin while the other player, Adam, tries to avoid the origin. The problem is to decide whether or not Eve has a winning strategy. In this paper we prove undecidability of the robot game in dimension two answering the question formulated by Doyen and Rabinovich in 2011 and closing the gap between undecidable and decidable cases.

Reino Niskanen, Igor Potapov, and Julien Reichert. Undecidability of Two-dimensional Robot Games. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 73:1-73:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{niskanen_et_al:LIPIcs.MFCS.2016.73, author = {Niskanen, Reino and Potapov, Igor and Reichert, Julien}, title = {{Undecidability of Two-dimensional Robot Games}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {73:1--73:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.73}, URN = {urn:nbn:de:0030-drops-64839}, doi = {10.4230/LIPIcs.MFCS.2016.73}, annote = {Keywords: reachability games, vector addition game, decidability, winning strategy} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

The decision problems on matrices were intensively studied for many decades as matrix products play an essential role in the representation of various computational processes. However, many computational problems for matrix semigroups are inherently difficult to solve even for problems in low dimensions and most matrix semigroup problems become undecidable in general starting from dimension three or four.
This paper solves two open problems about the decidability of the vector reachability problem over a finitely generated semigroup of matrices from SL(2, Z) and the point to point reachability (over rational numbers) for fractional linear transformations, where associated matrices are from SL(2, Z). The approach to solving reachability problems is based on the characterization of reachability paths between points which is followed by the translation of numerical problems on matrices into computational and combinatorial problems on words and formal languages. We also give a geometric interpretation of reachability paths and extend the decidability results to matrix products represented by arbitrary labelled directed graphs. Finally, we will use this technique to prove that a special case of the scalar reachability problem is decidable.

Igor Potapov and Pavel Semukhin. Vector Reachability Problem in SL(2, Z). In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 84:1-84:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{potapov_et_al:LIPIcs.MFCS.2016.84, author = {Potapov, Igor and Semukhin, Pavel}, title = {{Vector Reachability Problem in SL(2, Z)}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {84:1--84:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.84}, URN = {urn:nbn:de:0030-drops-64925}, doi = {10.4230/LIPIcs.MFCS.2016.84}, annote = {Keywords: matrix semigroup, vector reachability problem, special linear group, linear fractional transformation, automata and formal languages} }

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**Published in:** LIPIcs, Volume 24, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)

In this paper we investigate the decidability and complexity
of problems related to braid composition. While all known problems for a class of braids with 3 strands, B_3, have polynomial time solutions we prove that a very natural question for braid composition, the membership problem, is NP-hard for braids with only 3 strands. The membership problem is decidable for B_3, but it becomes harder for a class of braids with more strands. In particular we show that fundamental problems about braid compositions are undecidable for braids with at least 5 strands, but decidability of these problems for B_4 remains open. The paper introduces a few challenging algorithmic problems about topological braids opening new connections between braid groups, combinatorics on words, complexity theory and provides solutions for some of these problems by application of several techniques from automata theory, matrix semigroups and algorithms.

Igor Potapov. Composition Problems for Braids. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 24, pp. 175-187, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)

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@InProceedings{potapov:LIPIcs.FSTTCS.2013.175, author = {Potapov, Igor}, title = {{Composition Problems for Braids}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)}, pages = {175--187}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-64-4}, ISSN = {1868-8969}, year = {2013}, volume = {24}, editor = {Seth, Anil and Vishnoi, Nisheeth K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2013.175}, URN = {urn:nbn:de:0030-drops-43711}, doi = {10.4230/LIPIcs.FSTTCS.2013.175}, annote = {Keywords: Braid group, automata, group alphabet, combinatorics on words, matrix semigroups, NP-hardness, decidability} }

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