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**Published in:** LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

The betweenness centrality of a vertex v is an important centrality measure that quantifies how many optimal paths between pairs of other vertices visit v. Computing betweenness centrality in a temporal graph, in which the edge set may change over discrete timesteps, requires us to count temporal paths that are optimal with respect to some criterion. For several natural notions of optimality, including foremost or fastest temporal paths, this counting problem reduces to #TEMPORAL PATH, the problem of counting all temporal paths between a fixed pair of vertices; like the problems of counting foremost and fastest temporal paths, #TEMPORAL PATH is #P-hard in general. Motivated by the many applications of this intractable problem, we initiate a systematic study of the parameterised and approximation complexity of #TEMPORAL PATH. We show that the problem presumably does not admit an FPT-algorithm for the feedback vertex number of the static underlying graph, and that it is hard to approximate in general. On the positive side, we prove several exact and approximate FPT-algorithms for special cases.

Jessica Enright, Kitty Meeks, and Hendrik Molter. Counting Temporal Paths. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 30:1-30:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{enright_et_al:LIPIcs.STACS.2023.30, author = {Enright, Jessica and Meeks, Kitty and Molter, Hendrik}, title = {{Counting Temporal Paths}}, booktitle = {40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)}, pages = {30:1--30:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-266-2}, ISSN = {1868-8969}, year = {2023}, volume = {254}, editor = {Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.30}, URN = {urn:nbn:de:0030-drops-176829}, doi = {10.4230/LIPIcs.STACS.2023.30}, annote = {Keywords: Temporal Paths, Temporal Graphs, Parameterised Counting, Approximate Counting, #P-hard Counting Problems, Temporal Betweenness Centrality} }

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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

We study the problems of counting copies and induced copies of a small pattern graph H in a large host graph G. Recent work fully classified the complexity of those problems according to structural restrictions on the patterns H. In this work, we address the more challenging task of analysing the complexity for restricted patterns and restricted hosts. Specifically we ask which families of allowed patterns and hosts imply fixed-parameter tractability, i.e., the existence of an algorithm running in time f(H)⋅|G|^O(1) for some computable function f. Our main results present exhaustive and explicit complexity classifications for families that satisfy natural closure properties. Among others, we identify the problems of counting small matchings and independent sets in subgraph-closed graph classes 𝒢 as our central objects of study and establish the following crisp dichotomies as consequences of the Exponential Time Hypothesis:
- Counting k-matchings in a graph G ∈ 𝒢 is fixed-parameter tractable if and only if 𝒢 is nowhere dense.
- Counting k-independent sets in a graph G ∈ 𝒢 is fixed-parameter tractable if and only if 𝒢 is nowhere dense. Moreover, we obtain almost tight conditional lower bounds if 𝒢 is somewhere dense, i.e., not nowhere dense. These base cases of our classifications subsume a wide variety of previous results on the matching and independent set problem, such as counting k-matchings in bipartite graphs (Curticapean, Marx; FOCS 14), in F-colourable graphs (Roth, Wellnitz; SODA 20), and in degenerate graphs (Bressan, Roth; FOCS 21), as well as counting k-independent sets in bipartite graphs (Curticapean et al.; Algorithmica 19).
At the same time our proofs are much simpler: using structural characterisations of somewhere dense graphs, we show that a colourful version of a recent breakthrough technique for analysing pattern counting problems (Curticapean, Dell, Marx; STOC 17) applies to any subgraph-closed somewhere dense class of graphs, yielding a unified view of our current understanding of the complexity of subgraph counting.

Marco Bressan, Leslie Ann Goldberg, Kitty Meeks, and Marc Roth. Counting Subgraphs in Somewhere Dense Graphs. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 27:1-27:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bressan_et_al:LIPIcs.ITCS.2023.27, author = {Bressan, Marco and Goldberg, Leslie Ann and Meeks, Kitty and Roth, Marc}, title = {{Counting Subgraphs in Somewhere Dense Graphs}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {27:1--27:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.27}, URN = {urn:nbn:de:0030-drops-175304}, doi = {10.4230/LIPIcs.ITCS.2023.27}, annote = {Keywords: counting problems, somewhere dense graphs, parameterised complexity theory} }

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**Published in:** LIPIcs, Volume 226, 11th International Conference on Fun with Algorithms (FUN 2022)

It is well known that fighting a fire is a hard task. The Firefighter problem asks how to optimally deploy firefighters to defend the vertices of a graph from a fire. This problem is NP-Complete on all but a few classes of graphs. Thankfully, firefighters do not have to work alone, and are often aided by the efforts of good natured civillians who slow the spread of a fire by maintaining firebreaks when they are able. We will show that this help, although well-intentioned, unfortunately makes the optimal deployment of firefighters an even harder problem. To model this scenario we introduce the Temporal Firefighter problem, an extension of Firefighter to temporal graphs. We show that Temporal Firefighter is also NP-Complete, and remains so on all but one of the underlying classes of graphs on which Firefighter is known to have polynomial time solutions. This motivates us to explore making use of the temporal structure of the graph in our search for tractability, and we conclude by presenting an FPT algorithm for Temporal Firefighter with respect to the temporal graph parameter vertex-interval-membership-width.

Samuel D. Hand, Jessica Enright, and Kitty Meeks. Making Life More Confusing for Firefighters. In 11th International Conference on Fun with Algorithms (FUN 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 226, pp. 15:1-15:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{hand_et_al:LIPIcs.FUN.2022.15, author = {Hand, Samuel D. and Enright, Jessica and Meeks, Kitty}, title = {{Making Life More Confusing for Firefighters}}, booktitle = {11th International Conference on Fun with Algorithms (FUN 2022)}, pages = {15:1--15:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-232-7}, ISSN = {1868-8969}, year = {2022}, volume = {226}, editor = {Fraigniaud, Pierre and Uno, Yushi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2022.15}, URN = {urn:nbn:de:0030-drops-159851}, doi = {10.4230/LIPIcs.FUN.2022.15}, annote = {Keywords: Temporal graphs, Spreading processes, Parameterised complexity} }

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Brief Announcement

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

The Firefighter problem asks how many vertices can be saved from a fire spreading over the vertices of a graph. At timestep 0 a vertex begins burning, then on each subsequent timestep a non-burning vertex is chosen to be defended, and the fire then spreads to all undefended vertices that it neighbours. The problem is NP-Complete on arbitrary graphs, however existing work has found several graph classes for which there are polynomial time solutions. We introduce Temporal Firefighter, an extension of Firefighter to temporal graphs. We show that Temporal Firefighter is also NP-Complete, and remains so on all but one of the underlying classes of graphs on which Firefighter is known to have a polynomial-time solution. This motivates us to explore restrictions on the temporal structure of the graph, and we find that Temporal Firefighter is fixed parameter tractable with respect to the temporal graph parameter vertex-interval-membership-width.

Samuel D. Hand, Jessica Enright, and Kitty Meeks. Brief Announcement: The Temporal Firefighter Problem. In 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 221, pp. 22:1-22:3, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{hand_et_al:LIPIcs.SAND.2022.22, author = {Hand, Samuel D. and Enright, Jessica and Meeks, Kitty}, title = {{Brief Announcement: The Temporal Firefighter Problem}}, booktitle = {1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)}, pages = {22:1--22:3}, 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.2022.22}, URN = {urn:nbn:de:0030-drops-159644}, doi = {10.4230/LIPIcs.SAND.2022.22}, annote = {Keywords: Temporal graphs, Spreading processes, Parameterised complexity} }

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**Published in:** Dagstuhl Reports, Volume 11, Issue 3 (2021)

This report documents the program and the outcomes of Dagstuhl Seminar 121171 "Temporal Graphs: Structure, Algorithms, Applications". The seminar was organized around four core areas: models, concepts, classes; concrete algorithmic problems; distributed aspects; applications. Because of the ongoing pandemic crisis, the seminar had to be held fully online, with talk and open problems sessions focussing on afternoons. Besides 19 contributed talks and small-group discussions, there were lively open-problem sessions, and some of the problems and research directions proposed there are part of this document. Despite strongly missing the usual Dagstuhl atmosphere and personal interaction possibilities, the seminar helped to establish new contacts and to identify new research directions in a thriving research area between (algorithmic) graph theory and network science.

Arnaud Casteigts, Kitty Meeks, George B. Mertzios, and Rolf Niedermeier. Temporal Graphs: Structure, Algorithms, Applications (Dagstuhl Seminar 21171). In Dagstuhl Reports, Volume 11, Issue 3, pp. 16-46, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@Article{casteigts_et_al:DagRep.11.3.16, author = {Casteigts, Arnaud and Meeks, Kitty and Mertzios, George B. and Niedermeier, Rolf}, title = {{Temporal Graphs: Structure, Algorithms, Applications (Dagstuhl Seminar 21171)}}, pages = {16--46}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2021}, volume = {11}, number = {3}, editor = {Casteigts, Arnaud and Meeks, Kitty and Mertzios, George B. and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.11.3.16}, URN = {urn:nbn:de:0030-drops-146892}, doi = {10.4230/DagRep.11.3.16}, annote = {Keywords: algorithm engineering, complex network analysis, distributed computing, models and classes, parameterized complexity analysis} }

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

Spreading processes on graphs are a natural model for a wide variety of real-world phenomena, including information or behaviour spread over social networks, biological diseases spreading over contact or trade networks, and the potential flow of goods over logistical infrastructure. Often, the networks over which these processes spread are dynamic in nature, and can be modeled with graphs whose structure is subject to discrete changes over time, i.e. with temporal graphs. Here, we consider temporal graphs in which edges are available at specified timesteps, and study the problem of deleting edges from a given temporal graph in order to reduce the number of vertices (temporally) reachable from a given starting point. This could be used to control the spread of a disease, rumour, etc. in a temporal graph. In particular, our aim is to find a temporal subgraph in which a process starting at any single vertex can be transferred to only a limited number of other vertices using a temporally-feasible path (i.e. a path, along which the times of the edge availabilities increase). We introduce a natural deletion problem for temporal graphs and we provide positive and negative results on its computational complexity, both in the traditional and the parameterised sense (subject to various natural parameters), as well as addressing the approximability of this problem.

Jessica Enright, Kitty Meeks, George B. Mertzios, and Viktor Zamaraev. Deleting Edges to Restrict the Size of an Epidemic in Temporal Networks. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 57:1-57:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{enright_et_al:LIPIcs.MFCS.2019.57, author = {Enright, Jessica and Meeks, Kitty and Mertzios, George B. and Zamaraev, Viktor}, title = {{Deleting Edges to Restrict the Size of an Epidemic in Temporal Networks}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {57:1--57:15}, 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.2019.57}, URN = {urn:nbn:de:0030-drops-110010}, doi = {10.4230/LIPIcs.MFCS.2019.57}, annote = {Keywords: Temporal networks, spreading processes, graph modification, parameterised complexity} }

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**Published in:** LIPIcs, Volume 115, 13th International Symposium on Parameterized and Exact Computation (IPEC 2018)

The maximum modularity of a graph is a parameter widely used to describe the level of clustering or community structure in a network. Determining the maximum modularity of a graph is known to be NP-complete in general, and in practice a range of heuristics are used to construct partitions of the vertex-set which give lower bounds on the maximum modularity but without any guarantee on how close these bounds are to the true maximum. In this paper we investigate the parameterised complexity of determining the maximum modularity with respect to various standard structural parameterisations of the input graph G. We show that the problem belongs to FPT when parameterised by the size of a minimum vertex cover for G, and is solvable in polynomial time whenever the treewidth or max leaf number of G is bounded by some fixed constant; we also obtain an FPT algorithm, parameterised by treewidth, to compute any constant-factor approximation to the maximum modularity. On the other hand we show that the problem is W[1]-hard (and hence unlikely to admit an FPT algorithm) when parameterised simultaneously by pathwidth and the size of a minimum feedback vertex set.

Kitty Meeks and Fiona Skerman. The Parameterised Complexity of Computing the Maximum Modularity of a Graph. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 9:1-9:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{meeks_et_al:LIPIcs.IPEC.2018.9, author = {Meeks, Kitty and Skerman, Fiona}, title = {{The Parameterised Complexity of Computing the Maximum Modularity of a Graph}}, booktitle = {13th International Symposium on Parameterized and Exact Computation (IPEC 2018)}, pages = {9:1--9:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-084-2}, ISSN = {1868-8969}, year = {2019}, volume = {115}, editor = {Paul, Christophe and Pilipczuk, Michal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2018.9}, URN = {urn:nbn:de:0030-drops-102103}, doi = {10.4230/LIPIcs.IPEC.2018.9}, annote = {Keywords: modularity, community detection, integer quadratic programming, vertex cover, pathwidth} }

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**Published in:** LIPIcs, Volume 63, 11th International Symposium on Parameterized and Exact Computation (IPEC 2016)

Many combinatorial problems involve determining whether a universe of n elements contains a witness consisting of k elements which have some specified property. In this paper we investigate the relationship between the decision and enumeration versions of such problems: efficient methods are known for transforming a decision algorithm into a search procedure that finds a single witness, but even finding a second witness is not so straightforward in general. In this paper we show that, if the decision version of the problem belongs to FPT, there is a randomised algorithm which enumerates all witnesses in time f(k)*poly(n)*N, where N is the total number of witnesses and f is a computable function. This also gives rise to an efficient algorithm to count the total number of witnesses when this number is small.

Kitty Meeks. Randomised Enumeration of Small Witnesses Using a Decision Oracle. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 22:1-22:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{meeks:LIPIcs.IPEC.2016.22, author = {Meeks, Kitty}, title = {{Randomised Enumeration of Small Witnesses Using a Decision Oracle}}, booktitle = {11th International Symposium on Parameterized and Exact Computation (IPEC 2016)}, pages = {22:1--22:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-023-1}, ISSN = {1868-8969}, year = {2017}, volume = {63}, editor = {Guo, Jiong and Hermelin, Danny}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2016.22}, URN = {urn:nbn:de:0030-drops-69218}, doi = {10.4230/LIPIcs.IPEC.2016.22}, annote = {Keywords: enumeration algorithms, parameterized complexity, randomized algorithms, color coding} }

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