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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

A homomorphism from a graph G to a graph H is an edge-preserving mapping from V(G) to V(H). In the graph homomorphism problem, denoted by Hom(H), the graph H is fixed and we need to determine if there exists a homomorphism from an instance graph G to H. We study the complexity of the problem parameterized by the cutwidth of G, i.e., we assume that G is given along with a linear ordering v_1,…,v_n of V(G) such that, for each i ∈ {1,…,n-1}, the number of edges with one endpoint in {v_1,…,v_i} and the other in {v_{i+1},…,v_n} is at most k.
We aim, for each H, for algorithms for Hom(H) running in time c_H^k n^𝒪(1) and matching lower bounds that exclude c_H^{k⋅o(1)} n^𝒪(1) or c_H^{k(1-Ω(1))} n^𝒪(1) time algorithms under the (Strong) Exponential Time Hypothesis. In the paper we introduce a new parameter that we call mimsup(H). Our main contribution is strong evidence of a close connection between c_H and mimsup(H):
- an information-theoretic argument that the number of states needed in a natural dynamic programming algorithm is at most mimsup(H)^k,
- lower bounds that show that for almost all graphs H indeed we have c_H ≥ mimsup(H), assuming the (Strong) Exponential-Time Hypothesis, and
- an algorithm with running time exp(𝒪(mimsup(H)⋅k log k)) n^𝒪(1). In the last result we do not need to assume that H is a fixed graph. Thus, as a consequence, we obtain that the problem of deciding whether G admits a homomorphism to H is fixed-parameter tractable, when parameterized by cutwidth of G and mimsup(H).
The parameter mimsup(H) can be thought of as the p-th root of the maximum induced matching number in the graph obtained by multiplying p copies of H via a certain graph product, where p tends to infinity. It can also be defined as an asymptotic rank parameter of the adjacency matrix of H. Such parameters play a central role in, among others, algebraic complexity theory and additive combinatorics. Our results tightly link the parameterized complexity of a problem to such an asymptotic matrix parameter for the first time.

Carla Groenland, Isja Mannens, Jesper Nederlof, Marta Piecyk, and Paweł Rzążewski. Towards Tight Bounds for the Graph Homomorphism Problem Parameterized by Cutwidth via Asymptotic Matrix Parameters. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 77:1-77:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{groenland_et_al:LIPIcs.ICALP.2024.77, author = {Groenland, Carla and Mannens, Isja and Nederlof, Jesper and Piecyk, Marta and Rz\k{a}\.{z}ewski, Pawe{\l}}, title = {{Towards Tight Bounds for the Graph Homomorphism Problem Parameterized by Cutwidth via Asymptotic Matrix Parameters}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {77:1--77:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.77}, URN = {urn:nbn:de:0030-drops-202208}, doi = {10.4230/LIPIcs.ICALP.2024.77}, annote = {Keywords: graph homomorphism, cutwidth, asymptotic matrix parameters} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

We investigate the parameterized complexity of Binary CSP parameterized by the vertex cover number and the treedepth of the constraint graph, as well as by a selection of related modulator-based parameters. The main findings are as follows:
- Binary CSP parameterized by the vertex cover number is W[3]-complete. More generally, for every positive integer d, Binary CSP parameterized by the size of a modulator to a treedepth-d graph is W[2d+1]-complete. This provides a new family of natural problems that are complete for odd levels of the W-hierarchy.
- We introduce a new complexity class XSLP, defined so that Binary CSP parameterized by treedepth is complete for this class. We provide two equivalent characterizations of XSLP: the first one relates XSLP to a model of an alternating Turing machine with certain restrictions on conondeterminism and space complexity, while the second one links XSLP to the problem of model-checking first-order logic with suitably restricted universal quantification. Interestingly, the proof of the machine characterization of XSLP uses the concept of universal trees, which are prominently featured in the recent work on parity games.
- We describe a new complexity hierarchy sandwiched between the W-hierarchy and the A-hierarchy: For every odd t, we introduce a parameterized complexity class S[t] with W[t] ⊆ S[t] ⊆ A[t], defined using a parameter that interpolates between the vertex cover number and the treedepth. We expect that many of the studied classes will be useful in the future for pinpointing the complexity of various structural parameterizations of graph problems.

Hans L. Bodlaender, Carla Groenland, and Michał Pilipczuk. Parameterized Complexity of Binary CSP: Vertex Cover, Treedepth, and Related Parameters. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 27:1-27:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bodlaender_et_al:LIPIcs.ICALP.2023.27, author = {Bodlaender, Hans L. and Groenland, Carla and Pilipczuk, Micha{\l}}, title = {{Parameterized Complexity of Binary CSP: Vertex Cover, Treedepth, and Related Parameters}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {27:1--27:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.27}, URN = {urn:nbn:de:0030-drops-180798}, doi = {10.4230/LIPIcs.ICALP.2023.27}, annote = {Keywords: Parameterized Complexity, Constraint Satisfaction Problems, Binary CSP, List Coloring, Vertex Cover, Treedepth, W-hierarchy} }

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**Published in:** LIPIcs, Volume 249, 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)

In this paper, we introduce a new class of parameterized problems, which we call XALP: the class of all parameterized problems that can be solved in f(k)n^O(1) time and f(k)log n space on a non-deterministic Turing Machine with access to an auxiliary stack (with only top element lookup allowed). Various natural problems on "tree-structured graphs" are complete for this class: we show that List Coloring and All-or-Nothing Flow parameterized by treewidth are XALP-complete. Moreover, Independent Set and Dominating Set parameterized by treewidth divided by log n, and Max Cut parameterized by cliquewidth are also XALP-complete.
Besides finding a "natural home" for these problems, we also pave the road for future reductions. We give a number of equivalent characterisations of the class XALP, e.g., XALP is the class of problems solvable by an Alternating Turing Machine whose runs have tree size at most f(k)n^O(1) and use f(k)log n space. Moreover, we introduce "tree-shaped" variants of Weighted CNF-Satisfiability and Multicolor Clique that are XALP-complete.

Hans L. Bodlaender, Carla Groenland, Hugo Jacob, Marcin Pilipczuk, and Michał Pilipczuk. On the Complexity of Problems on Tree-Structured Graphs. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bodlaender_et_al:LIPIcs.IPEC.2022.6, author = {Bodlaender, Hans L. and Groenland, Carla and Jacob, Hugo and Pilipczuk, Marcin and Pilipczuk, Micha{\l}}, title = {{On the Complexity of Problems on Tree-Structured Graphs}}, booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)}, pages = {6:1--6:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-260-0}, ISSN = {1868-8969}, year = {2022}, volume = {249}, editor = {Dell, Holger and Nederlof, Jesper}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.6}, URN = {urn:nbn:de:0030-drops-173626}, doi = {10.4230/LIPIcs.IPEC.2022.6}, annote = {Keywords: Parameterized Complexity, Treewidth, XALP, XNLP} }

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**Published in:** LIPIcs, Volume 249, 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)

We study the parameterized complexity of computing the tree-partition-width, a graph parameter equivalent to treewidth on graphs of bounded maximum degree.
On one hand, we can obtain approximations of the tree-partition-width efficiently: we show that there is an algorithm that, given an n-vertex graph G and an integer k, constructs a tree-partition of width O(k⁷) for G or reports that G has tree-partition width more than k, in time k^O(1) n². We can improve slightly on the approximation factor by sacrificing the dependence on k, or on n.
On the other hand, we show the problem of computing tree-partition-width exactly is XALP-complete, which implies that it is W[t]-hard for all t. We deduce XALP-completeness of the problem of computing the domino treewidth. Finally, we adapt some known results on the parameter tree-partition-width and the topological minor relation, and use them to compare tree-partition-width to tree-cut width.

Hans L. Bodlaender, Carla Groenland, and Hugo Jacob. On the Parameterized Complexity of Computing Tree-Partitions. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bodlaender_et_al:LIPIcs.IPEC.2022.7, author = {Bodlaender, Hans L. and Groenland, Carla and Jacob, Hugo}, title = {{On the Parameterized Complexity of Computing Tree-Partitions}}, booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)}, pages = {7:1--7:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-260-0}, ISSN = {1868-8969}, year = {2022}, volume = {249}, editor = {Dell, Holger and Nederlof, Jesper}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.7}, URN = {urn:nbn:de:0030-drops-173633}, doi = {10.4230/LIPIcs.IPEC.2022.7}, annote = {Keywords: Parameterized algorithms, Tree partitions, tree-partition-width, Treewidth, Domino Treewidth, Approximation Algorithms, Parameterized Complexity} }

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**Published in:** LIPIcs, Volume 249, 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)

In this paper, we showcase the class XNLP as a natural place for many hard problems parameterized by linear width measures. This strengthens existing W[1]-hardness proofs for these problems, since XNLP-hardness implies W[t]-hardness for all t. It also indicates, via a conjecture by Pilipczuk and Wrochna [ToCT 2018], that any XP algorithm for such problems is likely to require XP space.
In particular, we show XNLP-completeness for natural problems parameterized by pathwidth, linear clique-width, and linear mim-width. The problems we consider are Independent Set, Dominating Set, Odd Cycle Transversal, (q-)Coloring, Max Cut, Maximum Regular Induced Subgraph, Feedback Vertex Set, Capacitated (Red-Blue) Dominating Set, and Bipartite Bandwidth.

Hans L. Bodlaender, Carla Groenland, Hugo Jacob, Lars Jaffke, and Paloma T. Lima. XNLP-Completeness for Parameterized Problems on Graphs with a Linear Structure. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bodlaender_et_al:LIPIcs.IPEC.2022.8, author = {Bodlaender, Hans L. and Groenland, Carla and Jacob, Hugo and Jaffke, Lars and Lima, Paloma T.}, title = {{XNLP-Completeness for Parameterized Problems on Graphs with a Linear Structure}}, booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)}, pages = {8:1--8:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-260-0}, ISSN = {1868-8969}, year = {2022}, volume = {249}, editor = {Dell, Holger and Nederlof, Jesper}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.8}, URN = {urn:nbn:de:0030-drops-173640}, doi = {10.4230/LIPIcs.IPEC.2022.8}, annote = {Keywords: parameterized complexity, XNLP, linear clique-width, W-hierarchy, pathwidth, linear mim-width, bandwidth} }

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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

We show that List Colouring can be solved on n-vertex trees by a deterministic Turing machine using O(log n) bits on the worktape. Given an n-vertex graph G = (V,E) and a list L(v) ⊆ {1,… ,n} of available colours for each v ∈ V, a list colouring for G is a proper colouring c such that c(v) ∈ L(v) for all v.

Hans L. Bodlaender, Carla Groenland, and Hugo Jacob. List Colouring Trees in Logarithmic Space. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bodlaender_et_al:LIPIcs.ESA.2022.24, author = {Bodlaender, Hans L. and Groenland, Carla and Jacob, Hugo}, title = {{List Colouring Trees in Logarithmic Space}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {24:1--24:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.24}, URN = {urn:nbn:de:0030-drops-169620}, doi = {10.4230/LIPIcs.ESA.2022.24}, annote = {Keywords: List colouring, trees, space complexity, logspace, graph algorithms, tree-partition-width} }

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**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

We study the fine-grained complexity of counting the number of colorings and connected spanning edge sets parameterized by the cutwidth and treewidth of the graph. While decompositions of small treewidth decompose the graph with small vertex separators, decompositions with small cutwidth decompose the graph with small edge separators.
Let p,q ∈ ℕ such that p is a prime and q ≥ 3. We show:
- If p divides q-1, there is a (q-1)^{ctw}n^{O(1)} time algorithm for counting list q-colorings modulo p of n-vertex graphs of cutwidth ctw. Furthermore, there is no ε > 0 for which there is a (q-1-ε)^{ctw} n^{O(1)} time algorithm that counts the number of list q-colorings modulo p of n-vertex graphs of cutwidth ctw, assuming the Strong Exponential Time Hypothesis (SETH).
- If p does not divide q-1, there is no ε > 0 for which there exists a (q-ε)^{ctw} n^{O(1)} time algorithm that counts the number of list q-colorings modulo p of n-vertex graphs of cutwidth ctw, assuming SETH. The lower bounds are in stark contrast with the existing 2^{ctw}n^{O(1)} time algorithm to compute the chromatic number of a graph by Jansen and Nederlof [Theor. Comput. Sci.'18].
Furthermore, by building upon the above lower bounds, we obtain the following lower bound for counting connected spanning edge sets: there is no ε > 0 for which there is an algorithm that, given a graph G and a cutwidth ordering of cutwidth ctw, counts the number of spanning connected edge sets of G modulo p in time (p - ε)^{ctw} n^{O(1)}, assuming SETH. We also give an algorithm with matching running time for this problem.
Before our work, even for the treewidth parameterization, the best conditional lower bound by Dell et al. [ACM Trans. Algorithms'14] only excluded 2^{o(tw)}n^{O(1)} time algorithms for this problem.
Both our algorithms and lower bounds employ use of the matrix rank method, by relating the complexity of the problem to the rank of a certain "compatibility matrix" in a non-trivial way.

Carla Groenland, Isja Mannens, Jesper Nederlof, and Krisztina Szilágyi. Tight Bounds for Counting Colorings and Connected Edge Sets Parameterized by Cutwidth. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 36:1-36:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{groenland_et_al:LIPIcs.STACS.2022.36, author = {Groenland, Carla and Mannens, Isja and Nederlof, Jesper and Szil\'{a}gyi, Krisztina}, title = {{Tight Bounds for Counting Colorings and Connected Edge Sets Parameterized by Cutwidth}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {36:1--36:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra 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.2022.36}, URN = {urn:nbn:de:0030-drops-158464}, doi = {10.4230/LIPIcs.STACS.2022.36}, annote = {Keywords: connected edge sets, cutwidth, parameterized algorithms, colorings, counting modulo p} }

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**Published in:** LIPIcs, Volume 214, 16th International Symposium on Parameterized and Exact Computation (IPEC 2021)

We settle the parameterized complexities of several variants of independent set reconfiguration and dominating set reconfiguration, parameterized by the number of tokens. We show that both problems are XL-complete when there is no limit on the number of moves and XNL-complete when a maximum length 𝓁 for the sequence is given in binary in the input. The problems are known to be XNLP-complete when 𝓁 is given in unary instead, and W[1]- and W[2]-hard respectively when 𝓁 is also a parameter. We complete the picture by showing membership in those classes.
Moreover, we show that for all the variants that we consider, token sliding and token jumping are equivalent under pl-reductions. We introduce partitioned variants of token jumping and token sliding, and give pl-reductions between the four variants that have precise control over the number of tokens and the length of the reconfiguration sequence.

Hans L. Bodlaender, Carla Groenland, and Céline M. F. Swennenhuis. Parameterized Complexities of Dominating and Independent Set Reconfiguration. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bodlaender_et_al:LIPIcs.IPEC.2021.9, author = {Bodlaender, Hans L. and Groenland, Carla and Swennenhuis, C\'{e}line M. F.}, title = {{Parameterized Complexities of Dominating and Independent Set Reconfiguration}}, booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)}, pages = {9:1--9:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-216-7}, ISSN = {1868-8969}, year = {2021}, volume = {214}, editor = {Golovach, Petr A. and Zehavi, Meirav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2021.9}, URN = {urn:nbn:de:0030-drops-153920}, doi = {10.4230/LIPIcs.IPEC.2021.9}, annote = {Keywords: Parameterized complexity, independent set reconfiguration, dominating set reconfiguration, W-hierarchy, XL, XNL, XNLP} }

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**Published in:** LIPIcs, Volume 209, 35th International Symposium on Distributed Computing (DISC 2021)

We show that there is a deterministic local algorithm (constant-time distributed graph algorithm) that finds a 5-approximation of a minimum dominating set on outerplanar graphs. We show there is no such algorithm that finds a (5-ε)-approximation, for any ε > 0. Our algorithm only requires knowledge of the degree of a vertex and of its neighbors, so that large messages and unique identifiers are not needed.

Marthe Bonamy, Linda Cook, Carla Groenland, and Alexandra Wesolek. A Tight Local Algorithm for the Minimum Dominating Set Problem in Outerplanar Graphs. In 35th International Symposium on Distributed Computing (DISC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 209, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bonamy_et_al:LIPIcs.DISC.2021.13, author = {Bonamy, Marthe and Cook, Linda and Groenland, Carla and Wesolek, Alexandra}, title = {{A Tight Local Algorithm for the Minimum Dominating Set Problem in Outerplanar Graphs}}, booktitle = {35th International Symposium on Distributed Computing (DISC 2021)}, pages = {13:1--13:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-210-5}, ISSN = {1868-8969}, year = {2021}, volume = {209}, editor = {Gilbert, Seth}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2021.13}, URN = {urn:nbn:de:0030-drops-148159}, doi = {10.4230/LIPIcs.DISC.2021.13}, annote = {Keywords: Outerplanar graphs, dominating set, LOCAL model, constant-factor approximation algorithm} }

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