Document

**Published in:** LIPIcs, Volume 292, 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)

Graph colouring is a fundamental problem in computer science, with a large body of research dedicated to both the general colouring problem and restricted cases. Harmonious colourings are one such restriction, where each edge must contain a globally unique pair of colours, i.e. if an edge connects a vertex coloured x with a vertex coloured y, then no other pair of connected vertices can be coloured x and y. Finding such a colouring in the traditional graph setting is known to be NP-hard, even in trees. This paper considers the generalisation of harmonious colourings to Temporal Graphs, specifically (k,t)-Temporal matchings, a class of temporal graphs where the underlying graph is a matching (a collection of disconnected components containing pairs of vertices), each edge can appear in at most t timesteps, and each timestep can contain at most k other edges. We provide a complete overview of the complexity landscape of finding temporal harmonious colourings for (k,t)-matchings. We show that finding a Temporal Harmonious Colouring, a colouring that is harmonious in each timestep, is NP-hard for (k,t)-Temporal Matchings when k ≥ 4, t ≥ 2, or when k ≥ 2 and t ≥ 3. We further show that this problem is inapproximable for t ≥ 2 and an unbounded value of k, and that the problem of determining the temporal harmonious chromatic number of a (2,3)-temporal matching can be determined in linear time. Finally, we strengthen this result by a set of upper and lower bounds of the temporal harmonious chromatic number both for individual temporal matchings and for the class of (k, t)-temporal matchings.

Duncan Adamson. Harmonious Colourings of Temporal Matchings. In 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 292, pp. 4:1-4:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{adamson:LIPIcs.SAND.2024.4, author = {Adamson, Duncan}, title = {{Harmonious Colourings of Temporal Matchings}}, booktitle = {3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)}, pages = {4:1--4:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-315-7}, ISSN = {1868-8969}, year = {2024}, volume = {292}, editor = {Casteigts, Arnaud and Kuhn, Fabian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2024.4}, URN = {urn:nbn:de:0030-drops-198823}, doi = {10.4230/LIPIcs.SAND.2024.4}, annote = {Keywords: Temporal Graphs, Harmonious Colouring, NP-Completeness} }

Document

Brief Announcement

**Published in:** LIPIcs, Volume 292, 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)

Robots are becoming an increasingly common part of scientific work within laboratory environments. In this paper, we investigate the problem of designing schedules for completing a set of tasks at fixed locations with multiple robots in a laboratory. We represent the laboratory as a graph with tasks placed on fixed vertices and robots represented as agents, with the constraint that no two robots may occupy the same vertex, or traverse the same edge, at the same time. Each schedule is partitioned into a set of timesteps, corresponding to a walk through the graph (allowing for a robot to wait at a vertex to complete a task), with each timestep taking time equal to the time for a robot to move from one vertex to another and each task taking some given number of timesteps during the completion of which a robot must stay at the vertex containing the task. The goal is to determine a set of schedules, with one schedule for each robot, minimising the number of timesteps taken by the schedule taking the greatest number of timesteps within the set of schedules.
We show that the problem of finding a task-fulfilling schedule in at most L timesteps is NP-complete for many simple classes of graphs. Explicitly, we provide this result for complete graphs, bipartite graphs, star graphs, and planar graphs. Finally, we provide positive results for line graphs, showing that we can find an optimal set of schedules for k robots completing m tasks of equal length of a path of length n in O(kmn) time, and a k-approximation when the length of the tasks is unbounded.

Duncan Adamson, Nathan Flaherty, Igor Potapov, and Paul G. Spirakis. Brief Announcement: Collision-Free Robot Scheduling. In 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 292, pp. 22:1-22:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{adamson_et_al:LIPIcs.SAND.2024.22, author = {Adamson, Duncan and Flaherty, Nathan and Potapov, Igor and Spirakis, Paul G.}, title = {{Brief Announcement: Collision-Free Robot Scheduling}}, booktitle = {3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)}, pages = {22:1--22:5}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-315-7}, ISSN = {1868-8969}, year = {2024}, volume = {292}, editor = {Casteigts, Arnaud and Kuhn, Fabian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2024.22}, URN = {urn:nbn:de:0030-drops-199004}, doi = {10.4230/LIPIcs.SAND.2024.22}, annote = {Keywords: Graph Exploration, Scheduling, NP-Completeness, Approximation Algorithms} }

Document

**Published in:** LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

A subsequence of a word w is a word u such that u = w[i₁] w[i₂] … w[i_k], for some set of indices 1 ≤ i₁ < i₂ < … < i_k ≤ |w|. A word w is k-subsequence universal over an alphabet Σ if every word in Σ^k appears in w as a subsequence. In this paper, we study the intersection between the set of k-subsequence universal words over some alphabet Σ and regular languages over Σ. We call a regular language L k-∃-subsequence universal if there exists a k-subsequence universal word in L, and k-∀-subsequence universal if every word of L is k-subsequence universal. We give algorithms solving the problems of deciding if a given regular language, represented by a finite automaton recognising it, is k-∃-subsequence universal and, respectively, if it is k-∀-subsequence universal, for a given k. The algorithms are FPT w.r.t. the size of the input alphabet, and their run-time does not depend on k; they run in polynomial time in the number n of states of the input automaton when the size of the input alphabet is O(log n). Moreover, we show that the problem of deciding if a given regular language is k-∃-subsequence universal is NP-complete, when the language is over a large alphabet. Further, we provide algorithms for counting the number of k-subsequence universal words (paths) accepted by a given deterministic (respectively, nondeterministic) finite automaton, and ranking an input word (path) within the set of k-subsequence universal words accepted by a given finite automaton.

Duncan Adamson, Pamela Fleischmann, Annika Huch, Tore Koß, Florin Manea, and Dirk Nowotka. k-Universality of Regular Languages. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 4:1-4:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{adamson_et_al:LIPIcs.ISAAC.2023.4, author = {Adamson, Duncan and Fleischmann, Pamela and Huch, Annika and Ko{\ss}, Tore and Manea, Florin and Nowotka, Dirk}, title = {{k-Universality of Regular Languages}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {4:1--4:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-289-1}, ISSN = {1868-8969}, year = {2023}, volume = {283}, editor = {Iwata, Satoru and Kakimura, Naonori}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.4}, URN = {urn:nbn:de:0030-drops-193064}, doi = {10.4230/LIPIcs.ISAAC.2023.4}, annote = {Keywords: String Algorithms, Regular Languages, Finite Automata, Subsequences} }

Document

**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)

Copy BibTex To Clipboard

@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} }

Document

**Published in:** LIPIcs, Volume 221, 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)

A temporal graph G = (G_1, G_2, ..., G_T) is a graph represented by a sequence of T graphs over a common set of vertices, such that at the i-th time step only the edge set E_i is active. The temporal graph exploration problem asks for a shortest temporal walk on some temporal graph visiting every vertex. We show that temporal graphs with n vertices can be explored in O(k n^{1.5} log n) days if the underlying graph has treewidth k and in O(n^{1.75} log n) days if the underlying graph is planar. Furthermore, we show that any temporal graph whose underlying graph is a cycle with k chords can be explored in at most 6kn days. Finally, we demonstrate that there are temporal realisations of sub cubic planar graphs that cannot be explored faster than in Ω(n log n) days. All these improve best known results in the literature.

Duncan Adamson, Vladimir V. Gusev, Dmitriy Malyshev, and Viktor Zamaraev. Faster Exploration of Some Temporal Graphs. In 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 221, pp. 5:1-5:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{adamson_et_al:LIPIcs.SAND.2022.5, author = {Adamson, Duncan and Gusev, Vladimir V. and Malyshev, Dmitriy and Zamaraev, Viktor}, title = {{Faster Exploration of Some Temporal Graphs}}, booktitle = {1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)}, pages = {5:1--5:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-224-2}, ISSN = {1868-8969}, year = {2022}, volume = {221}, editor = {Aspnes, James and Michail, Othon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2022.5}, URN = {urn:nbn:de:0030-drops-159475}, doi = {10.4230/LIPIcs.SAND.2022.5}, annote = {Keywords: Temporal Graphs, Graph Exploration} }

Document

**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)

Copy BibTex To Clipboard

@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} }

X

Feedback for Dagstuhl Publishing

Feedback submitted

Please try again later or send an E-mail