Document

**Published in:** Dagstuhl Reports, Volume 13, Issue 8 (2024)

This report documents the program and the outcomes of Dagstuhl Seminar 23331 "Recent Trends in Graph Decomposition", which took place from 13. August to 18. August, 2023. The seminar brought together 33 experts from academia and industry to discuss graph decomposition, a pivotal technique for handling massive graphs in applications such as social networks and scientific simulations. The seminar addressed the challenges posed by contemporary hardware designs, the potential of deep neural networks and reinforcement learning in developing heuristics, the unique optimization requirements of large sparse data, and the need for scalable algorithms suitable for emerging architectures. Through presentations, discussions, and collaborative sessions, the event fostered an exchange of innovative ideas, leading to the creation of community notes highlighting key open problems in the field.

George Karypis, Christian Schulz, Darren Strash, Deepak Ajwani, Rob H. Bisseling, Katrin Casel, Ümit V. Çatalyürek, Cédric Chevalier, Florian Chudigiewitsch, Marcelo Fonseca Faraj, Michael Fellows, Lars Gottesbüren, Tobias Heuer, Kamer Kaya, Jakub Lacki, Johannes Langguth, Xiaoye Sherry Li, Ruben Mayer, Johannes Meintrup, Yosuke Mizutani, François Pellegrini, Fabrizio Petrini, Frances Rosamond, Ilya Safro, Sebastian Schlag, Roohani Sharma, Blair D. Sullivan, Bora Uçar, and Albert-Jan Yzelman. Recent Trends in Graph Decomposition (Dagstuhl Seminar 23331). In Dagstuhl Reports, Volume 13, Issue 8, pp. 1-45, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@Article{karypis_et_al:DagRep.13.8.1, author = {Karypis, George and Schulz, Christian and Strash, Darren and Ajwani, Deepak and Bisseling, Rob H. and Casel, Katrin and \c{C}ataly\"{u}rek, \"{U}mit V. and Chevalier, C\'{e}dric and Chudigiewitsch, Florian and Faraj, Marcelo Fonseca and Fellows, Michael and Gottesb\"{u}ren, Lars and Heuer, Tobias and Kaya, Kamer and Lacki, Jakub and Langguth, Johannes and Li, Xiaoye Sherry and Mayer, Ruben and Meintrup, Johannes and Mizutani, Yosuke and Pellegrini, Fran\c{c}ois and Petrini, Fabrizio and Rosamond, Frances and Safro, Ilya and Schlag, Sebastian and Sharma, Roohani and Sullivan, Blair D. and U\c{c}ar, Bora and Yzelman, Albert-Jan}, title = {{Recent Trends in Graph Decomposition (Dagstuhl Seminar 23331)}}, pages = {1--45}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2024}, volume = {13}, number = {8}, editor = {Karypis, George and Schulz, Christian and Strash, Darren}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.13.8.1}, URN = {urn:nbn:de:0030-drops-198114}, doi = {10.4230/DagRep.13.8.1}, annote = {Keywords: combinatorial optimization, experimental algorithmics, parallel algorithms} }

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**Published in:** LIPIcs, Volume 265, 21st International Symposium on Experimental Algorithms (SEA 2023)

In the Directed Feedback Vertex Set (DFVS) problem, one is given a directed graph G = (V,E) and wants to find a minimum cardinality set S ⊆ V such that G-S is acyclic. DFVS is a fundamental problem in computer science and finds applications in areas such as deadlock detection. The problem was the subject of the 2022 PACE coding challenge. We develop a novel exact algorithm for the problem that is tailored to perform well on instances that are mostly bi-directed. For such instances, we adapt techniques from the well-researched vertex cover problem. Our core idea is an iterative reduction to vertex cover. To this end, we also develop a new reduction rule that reduces the number of not bi-directed edges. With the resulting algorithm, we were able to win third place in the exact track of the PACE challenge. We perform computational experiments and compare the running time to other exact algorithms, in particular to the winning algorithm in PACE. Our experiments show that we outpace the other algorithms on instances that have a low density of uni-directed edges.

Sebastian Angrick, Ben Bals, Katrin Casel, Sarel Cohen, Tobias Friedrich, Niko Hastrich, Theresa Hradilak, Davis Issac, Otto Kißig, Jonas Schmidt, and Leo Wendt. Solving Directed Feedback Vertex Set by Iterative Reduction to Vertex Cover. In 21st International Symposium on Experimental Algorithms (SEA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 265, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{angrick_et_al:LIPIcs.SEA.2023.10, author = {Angrick, Sebastian and Bals, Ben and Casel, Katrin and Cohen, Sarel and Friedrich, Tobias and Hastrich, Niko and Hradilak, Theresa and Issac, Davis and Ki{\ss}ig, Otto and Schmidt, Jonas and Wendt, Leo}, title = {{Solving Directed Feedback Vertex Set by Iterative Reduction to Vertex Cover}}, booktitle = {21st International Symposium on Experimental Algorithms (SEA 2023)}, pages = {10:1--10:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-279-2}, ISSN = {1868-8969}, year = {2023}, volume = {265}, editor = {Georgiadis, Loukas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2023.10}, URN = {urn:nbn:de:0030-drops-183602}, doi = {10.4230/LIPIcs.SEA.2023.10}, annote = {Keywords: directed feedback vertex set, vertex cover, reduction rules} }

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**Published in:** LIPIcs, Volume 256, 4th Symposium on Foundations of Responsible Computing (FORC 2023)

The study of algorithmic fairness received growing attention recently. This stems from the awareness that bias in the input data for machine learning systems may result in discriminatory outputs. For clustering tasks, one of the most central notions of fairness is the formalization by Chierichetti, Kumar, Lattanzi, and Vassilvitskii [NeurIPS 2017]. A clustering is said to be fair, if each cluster has the same distribution of manifestations of a sensitive attribute as the whole input set. This is motivated by various applications where the objects to be clustered have sensitive attributes that should not be over- or underrepresented. Most research on this version of fair clustering has focused on centriod-based objectives.
In contrast, we discuss the applicability of this fairness notion to Correlation Clustering. The existing literature on the resulting Fair Correlation Clustering problem either presents approximation algorithms with poor approximation guarantees or severely limits the possible distributions of the sensitive attribute (often only two manifestations with a 1:1 ratio are considered). Our goal is to understand if there is hope for better results in between these two extremes. To this end, we consider restricted graph classes which allow us to characterize the distributions of sensitive attributes for which this form of fairness is tractable from a complexity point of view.
While existing work on Fair Correlation Clustering gives approximation algorithms, we focus on exact solutions and investigate whether there are efficiently solvable instances. The unfair version of Correlation Clustering is trivial on forests, but adding fairness creates a surprisingly rich picture of complexities. We give an overview of the distributions and types of forests where Fair Correlation Clustering turns from tractable to intractable.
As the most surprising insight, we consider the fact that the cause of the hardness of Fair Correlation Clustering is not the strictness of the fairness condition. We lift most of our results to also hold for the relaxed version of the fairness condition. Instead, the source of hardness seems to be the distribution of the sensitive attribute. On the positive side, we identify some reasonable distributions that are indeed tractable. While this tractability is only shown for forests, it may open an avenue to design reasonable approximations for larger graph classes.

Katrin Casel, Tobias Friedrich, Martin Schirneck, and Simon Wietheger. Fair Correlation Clustering in Forests. In 4th Symposium on Foundations of Responsible Computing (FORC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 256, pp. 9:1-9:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{casel_et_al:LIPIcs.FORC.2023.9, author = {Casel, Katrin and Friedrich, Tobias and Schirneck, Martin and Wietheger, Simon}, title = {{Fair Correlation Clustering in Forests}}, booktitle = {4th Symposium on Foundations of Responsible Computing (FORC 2023)}, pages = {9:1--9:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-272-3}, ISSN = {1868-8969}, year = {2023}, volume = {256}, editor = {Talwar, Kunal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FORC.2023.9}, URN = {urn:nbn:de:0030-drops-179307}, doi = {10.4230/LIPIcs.FORC.2023.9}, annote = {Keywords: correlation clustering, disparate impact, fair clustering, relaxed fairness} }

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PACE Solver Description

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

In this document we describe the techniques we used and implemented for our submission to the Parameterized Algorithms and Computational Experiments Challenge (PACE) 2022. The given problem is Directed Feedback Vertex Set (DFVS), where one is given a directed graph G = (V,E) and wants to find a minimum S ⊆ V such that G-S is acyclic. We approach this problem by first exhaustively applying a set of reduction rules. In order to find a minimum DFVS on the remaining instance, we create and solve a series of Vertex Cover instances.

Sebastian Angrick, Ben Bals, Katrin Casel, Sarel Cohen, Tobias Friedrich, Niko Hastrich, Theresa Hradilak, Davis Issac, Otto Kißig, Jonas Schmidt, and Leo Wendt. PACE Solver Description: Mount Doom - An Exact Solver for Directed Feedback Vertex Set. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 28:1-28:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{angrick_et_al:LIPIcs.IPEC.2022.28, author = {Angrick, Sebastian and Bals, Ben and Casel, Katrin and Cohen, Sarel and Friedrich, Tobias and Hastrich, Niko and Hradilak, Theresa and Issac, Davis and Ki{\ss}ig, Otto and Schmidt, Jonas and Wendt, Leo}, title = {{PACE Solver Description: Mount Doom - An Exact Solver for Directed Feedback Vertex Set}}, booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)}, pages = {28:1--28:4}, 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.2022.28}, URN = {urn:nbn:de:0030-drops-173847}, doi = {10.4230/LIPIcs.IPEC.2022.28}, annote = {Keywords: directed feedback vertex set, vertex cover, reduction rules} }

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**Published in:** LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)

We consider the problem of partitioning a graph into a non-fixed number of non-overlapping subgraphs of maximum density. The density of a partition is the sum of the densities of the subgraphs, where the density of a subgraph is half its average degree, that is, the ratio of its number of edges and its number of vertices. This problem, called Dense Graph Partition, is known to be NP-hard on general graphs and polynomial-time solvable on trees, and polynomial-time 2-approximable.
In this paper we study the restriction of Dense Graph Partition to particular sparse and dense graph classes. In particular, we prove that it is NP-hard on dense bipartite graphs as well as on cubic graphs. On dense graphs on n vertices, it is polynomial-time solvable on graphs with minimum degree n-3 and NP-hard on (n-4)-regular graphs. We prove that it is polynomial-time 4/3-approximable on cubic graphs and admits an efficient polynomial-time approximation scheme on graphs of minimum degree n-t for any constant t ≥ 4.

Cristina Bazgan, Katrin Casel, and Pierre Cazals. Dense Graph Partitioning on Sparse and Dense Graphs. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bazgan_et_al:LIPIcs.SWAT.2022.13, author = {Bazgan, Cristina and Casel, Katrin and Cazals, Pierre}, title = {{Dense Graph Partitioning on Sparse and Dense Graphs}}, booktitle = {18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)}, pages = {13:1--13:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-236-5}, ISSN = {1868-8969}, year = {2022}, volume = {227}, editor = {Czumaj, Artur and Xin, Qin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.13}, URN = {urn:nbn:de:0030-drops-161732}, doi = {10.4230/LIPIcs.SWAT.2022.13}, annote = {Keywords: NP-hardness, approximation, density, graph partitioning, bipartite graphs, cubic graphs, dense graphs} }

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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

We combine ideas from distance sensitivity oracles (DSOs) and fixed-parameter tractability (FPT) to design sensitivity oracles for FPT graph problems. An oracle with sensitivity f for an FPT problem Π on a graph G with parameter k preprocesses G in time O(g(f,k) ⋅ poly(n)). When queried with a set F of at most f edges of G, the oracle reports the answer to the Π - with the same parameter k - on the graph G-F, i.e., G deprived of F. The oracle should answer queries in a time that is significantly faster than merely running the best-known FPT algorithm on G-F from scratch.
We design sensitivity oracles for the k-Path and the k-Vertex Cover problem. Our first oracle for k-Path has size O(k^{f+1}) and query time O(f min{f, log(f) + k}). We use a technique inspired by the work of Weimann and Yuster [FOCS 2010, TALG 2013] on distance sensitivity problems to reduce the space to O(({f+k}/f)^f ({f+k}/k)^k fk⋅log(n)) at the expense of increasing the query time to O(({f+k}/f)^f ({f+k}/k)^k f min{f,k}⋅log(n)). Both oracles can be modified to handle vertex-failures, but we need to replace k with 2k in all the claimed bounds.
Regarding k-Vertex Cover, we design three oracles offering different trade-offs between the size and the query time. The first oracle takes O(3^{f+k}) space and has O(2^f) query time, the second one has a size of O(2^{f+k²+k}) and a query time of O(f+k²); finally, the third one takes O(fk+k²) space and can be queried in time O(1.2738^k + f). All our oracles are computable in time (at most) proportional to their size and the time needed to detect a k-path or k-vertex cover, respectively. We also provide an interesting connection between k-Vertex Cover and the fault-tolerant shortest path problem, by giving a DSO of size O(poly(f,k) ⋅ n) with query time in O(poly(f,k)), where k is the size of a vertex cover.
Following our line of research connecting fault-tolerant FPT and shortest paths problems, we introduce parameterization to the computation of distance preservers. We study the problem, given a directed unweighted graph with a fixed source s and parameters f and k, to construct a polynomial-sized oracle that efficiently reports, for any target vertex v and set F of at most f edges, whether the distance from s to v increases at most by an additive term of k in G-F. The oracle size is O(2^k k²⋅n), while the time needed to answer a query is O(2^k f^ω k^ω), where ω < 2.373 is the matrix multiplication exponent. The second problem we study is about the construction of bounded-stretch fault-tolerant preservers. We construct a subgraph with O(2^{fk+f+k} k ⋅ n) edges that preserves those s-v-distances that do not increase by more than k upon failure of F. This improves significantly over the Õ(f n^{2-1/(2^f)}) bound in the unparameterized case by Bodwin et al. [ICALP 2017].

Davide Bilò, Katrin Casel, Keerti Choudhary, Sarel Cohen, Tobias Friedrich, J.A. Gregor Lagodzinski, Martin Schirneck, and Simon Wietheger. Fixed-Parameter Sensitivity Oracles. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bilo_et_al:LIPIcs.ITCS.2022.23, author = {Bil\`{o}, Davide and Casel, Katrin and Choudhary, Keerti and Cohen, Sarel and Friedrich, Tobias and Lagodzinski, J.A. Gregor and Schirneck, Martin and Wietheger, Simon}, title = {{Fixed-Parameter Sensitivity Oracles}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {23:1--23:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.23}, URN = {urn:nbn:de:0030-drops-156196}, doi = {10.4230/LIPIcs.ITCS.2022.23}, annote = {Keywords: data structures, distance preservers, distance sensitivity oracles, fault tolerance, fixed-parameter tractability, k-path, vertex cover} }

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APPROX

**Published in:** LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

A connected partition is a partition of the vertices of a graph into sets that induce connected subgraphs. Such partitions naturally occur in many application areas such as road networks, and image processing. In these settings, it is often desirable to partition into a fixed number of parts of roughly of the same size or weight. The resulting computational problem is called Balanced Connected Partition (BCP). The two classical objectives for BCP are to maximize the weight of the smallest, or minimize the weight of the largest component. We study BCP on c-claw-free graphs, the class of graphs that do not have K_{1,c} as an induced subgraph, and present efficient (c-1)-approximation algorithms for both objectives. In particular, for 3-claw-free graphs, also simply known as claw-free graphs, we obtain a 2-approximation. Due to the claw-freeness of line graphs, this also implies a 2-approximation for the edge-partition version of BCP in general graphs.
A harder connected partition problem arises from demanding a connected partition into k parts that have (possibly) heterogeneous target weights w₁,…,w_k. In the 1970s Győri and Lovász showed that if G is k-connected and the target weights sum to the total size of G, such a partition exists. However, to this day no polynomial algorithm to compute such partitions exists for k > 4. Towards finding such a partition T₁,…, T_k in k-connected graphs for general k, we show how to efficiently compute connected partitions that at least approximately meet the target weights, subject to the mild assumption that each w_i is greater than the weight of the heaviest vertex. In particular, we give a 3-approximation for both the lower and the upper bounded version i.e. we guarantee that each T_i has weight at least (w_i)/3 or that each T_i has weight most 3w_i, respectively. Also, we present a both-side bounded version that produces a connected partition where each T_i has size at least (w_i)/3 and at most max({r,3}) w_i, where r ≥ 1 is the ratio between the largest and smallest value in w₁, … , w_k. In particular for the balanced version, i.e. w₁ = w₂ = , … , = w_k, this gives a partition with 1/3w_i ≤ w(T_i) ≤ 3w_i.

Ralf Borndörfer, Katrin Casel, Davis Issac, Aikaterini Niklanovits, Stephan Schwartz, and Ziena Zeif. Connected k-Partition of k-Connected Graphs and c-Claw-Free Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{borndorfer_et_al:LIPIcs.APPROX/RANDOM.2021.27, author = {Bornd\"{o}rfer, Ralf and Casel, Katrin and Issac, Davis and Niklanovits, Aikaterini and Schwartz, Stephan and Zeif, Ziena}, title = {{Connected k-Partition of k-Connected Graphs and c-Claw-Free Graphs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)}, pages = {27:1--27:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-207-5}, ISSN = {1868-8969}, year = {2021}, volume = {207}, editor = {Wootters, Mary and Sanit\`{a}, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.27}, URN = {urn:nbn:de:0030-drops-147200}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.27}, annote = {Keywords: connected partition, Gy\H{o}ri-Lov\'{a}sz, balanced partition, approximation algorithms} }

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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

We introduce the balanced crown decomposition that captures the structure imposed on graphs by their connected induced subgraphs of a given size. Such subgraphs are a popular modeling tool in various application areas, where the non-local nature of the connectivity condition usually results in very challenging algorithmic tasks. The balanced crown decomposition is a combination of a crown decomposition and a balanced partition which makes it applicable to graph editing as well as graph packing and partitioning problems. We illustrate this by deriving improved approximation algorithms and kernelization for a variety of such problems.
In particular, through this structure, we obtain the first constant-factor approximation for the Balanced Connected Partition (BCP) problem, where the task is to partition a vertex-weighted graph into k connected components of approximately equal weight. We derive a 3-approximation for the two most commonly used objectives of maximizing the weight of the lightest component or minimizing the weight of the heaviest component.

Katrin Casel, Tobias Friedrich, Davis Issac, Aikaterini Niklanovits, and Ziena Zeif. Balanced Crown Decomposition for Connectivity Constraints. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{casel_et_al:LIPIcs.ESA.2021.26, author = {Casel, Katrin and Friedrich, Tobias and Issac, Davis and Niklanovits, Aikaterini and Zeif, Ziena}, title = {{Balanced Crown Decomposition for Connectivity Constraints}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {26:1--26:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.26}, URN = {urn:nbn:de:0030-drops-146076}, doi = {10.4230/LIPIcs.ESA.2021.26}, annote = {Keywords: crown decomposition, connected partition, balanced partition, approximation algorithms} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

We study the problem of counting the number of homomorphisms from an input graph G to a fixed (quantum) graph ̄{H} in any finite field of prime order ℤ_p. The subproblem with graph H was introduced by Faben and Jerrum [ToC'15] and its complexity is still uncharacterised despite active research, e.g. the very recent work of Focke, Goldberg, Roth, and Zivný [SODA'21]. Our contribution is threefold.
First, we introduce the study of quantum graphs to the study of modular counting homomorphisms. We show that the complexity for a quantum graph ̄{H} collapses to the complexity criteria found at dimension 1: graphs. Second, in order to prove cases of intractability we establish a further reduction to the study of bipartite graphs. Lastly, we establish a dichotomy for all bipartite (K_{3,3}$1{e}, {domino})-free graphs by a thorough structural study incorporating both local and global arguments. This result subsumes all results on bipartite graphs known for all prime moduli and extends them significantly. Even for the subproblem with p = 2 this establishes new results.

J. A. Gregor Lagodzinski, Andreas Göbel, Katrin Casel, and Tobias Friedrich. On Counting (Quantum-)Graph Homomorphisms in Finite Fields of Prime Order. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 91:1-91:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{lagodzinski_et_al:LIPIcs.ICALP.2021.91, author = {Lagodzinski, J. A. Gregor and G\"{o}bel, Andreas and Casel, Katrin and Friedrich, Tobias}, title = {{On Counting (Quantum-)Graph Homomorphisms in Finite Fields of Prime Order}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {91:1--91:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela 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.ICALP.2021.91}, URN = {urn:nbn:de:0030-drops-141608}, doi = {10.4230/LIPIcs.ICALP.2021.91}, annote = {Keywords: Algorithms, Theory, Quantum Graphs, Bipartite Graphs, Graph Homomorphisms, Modular Counting, Complexity Dichotomy} }

Document

**Published in:** LIPIcs, Volume 186, 24th International Conference on Database Theory (ICDT 2021)

A regular path query (RPQ) is a regular expression q that returns all node pairs (u, v) from a graph database that are connected by an arbitrary path labelled with a word from L(q). The obvious algorithmic approach to RPQ evaluation (called PG-approach), i. e., constructing the product graph between an NFA for q and the graph database, is appealing due to its simplicity and also leads to efficient algorithms. However, it is unclear whether the PG-approach is optimal. We address this question by thoroughly investigating which upper complexity bounds can be achieved by the PG-approach, and we complement these with conditional lower bounds (in the sense of the fine-grained complexity framework). A special focus is put on enumeration and delay bounds, as well as the data complexity perspective. A main insight is that we can achieve optimal (or near optimal) algorithms with the PG-approach, but the delay for enumeration is rather high (linear in the database). We explore three successful approaches towards enumeration with sub-linear delay: super-linear preprocessing, approximations of the solution sets, and restricted classes of RPQs.

Katrin Casel and Markus L. Schmid. Fine-Grained Complexity of Regular Path Queries. In 24th International Conference on Database Theory (ICDT 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 186, pp. 19:1-19:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{casel_et_al:LIPIcs.ICDT.2021.19, author = {Casel, Katrin and Schmid, Markus L.}, title = {{Fine-Grained Complexity of Regular Path Queries}}, booktitle = {24th International Conference on Database Theory (ICDT 2021)}, pages = {19:1--19:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-179-5}, ISSN = {1868-8969}, year = {2021}, volume = {186}, editor = {Yi, Ke and Wei, Zhewei}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2021.19}, URN = {urn:nbn:de:0030-drops-137277}, doi = {10.4230/LIPIcs.ICDT.2021.19}, annote = {Keywords: Graph Databases, Regular Path Queries, Enumeration, Fine-Grained Complexity} }

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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 115, 13th International Symposium on Parameterized and Exact Computation (IPEC 2018)

This paper considers the effect of non-metric distances for lower-bounded clustering, i.e., the problem of computing a partition for a given set of objects with pairwise distance, such that each set has a certain minimum cardinality (as required for anonymisation or balanced facility location problems). We discuss lower-bounded clustering with the objective to minimise the maximum radius or diameter of the clusters. For these problems there exists a 2-approximation but only if the pairwise distance on the objects satisfies the triangle inequality, without this property no polynomial-time constant factor approximation is possible, unless P=NP. We try to resolve or at least soften this effect of non-metric distances by devising particular strategies to deal with violations of the triangle inequality (conflicts). With parameterised algorithmics, we find that if the number of such conflicts is not too large, constant factor approximations can still be computed efficiently.
In particular, we introduce parameterised approximations with respect to not just the number of conflicts but also for the vertex cover number of the conflict graph (graph induced by conflicts). Interestingly, we salvage the approximation ratio of 2 for diameter while for radius it is only possible to show a ratio of 3. For the parameter vertex cover number of the conflict graph this worsening in ratio is shown to be unavoidable, unless FPT=W[2]. We further discuss improvements for diameter by choosing the (induced) P_3-cover number of the conflict graph as parameter and complement these by showing that, unless FPT=W[1], there exists no constant factor parameterised approximation with respect to the parameter split vertex deletion set.

Katrin Casel. Resolving Conflicts for Lower-Bounded Clustering. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 23:1-23:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{casel:LIPIcs.IPEC.2018.23, author = {Casel, Katrin}, title = {{Resolving Conflicts for Lower-Bounded Clustering}}, booktitle = {13th International Symposium on Parameterized and Exact Computation (IPEC 2018)}, pages = {23:1--23:14}, 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.2018.23}, URN = {urn:nbn:de:0030-drops-102247}, doi = {10.4230/LIPIcs.IPEC.2018.23}, annote = {Keywords: clustering, triangle inequality, parameterised approximation} }

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

Unit square (grid) visibility graphs (USV and USGV, resp.) are described by axis-parallel visibility between unit squares placed (on integer grid coordinates) in the plane. We investigate combinatorial properties of these graph classes and the hardness of variants of the recognition problem, i.e., the problem of representing USGV with fixed visibilities within small area and, for USV, the general recognition problem.

Katrin Casel, Henning Fernau, Alexander Grigoriev, Markus L. Schmid, and Sue Whitesides. Combinatorial Properties and Recognition of Unit Square Visibility Graphs. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{casel_et_al:LIPIcs.MFCS.2017.30, author = {Casel, Katrin and Fernau, Henning and Grigoriev, Alexander and Schmid, Markus L. and Whitesides, Sue}, title = {{Combinatorial Properties and Recognition of Unit Square Visibility Graphs}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {30:1--30:15}, 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.2017.30}, URN = {urn:nbn:de:0030-drops-80770}, doi = {10.4230/LIPIcs.MFCS.2017.30}, annote = {Keywords: Visibility graphs, visibility layout, NP-completeness, exact algorithms} }

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**Published in:** LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

Classical clustering problems search for a partition of objects into a fixed number of clusters. In many scenarios however the number of clusters is not known or necessarily fixed. Further, clusters are sometimes only considered to be of significance if they have a certain size. We discuss clustering into sets of minimum cardinality k without a fixed number of sets and present a general model for these types of problems. This general framework allows the comparison of different measures to assess the quality of a clustering. We specifically consider nine quality-measures and classify the complexity of the resulting problems with respect to k. Further, we derive some polynomial-time solvable cases for k = 2 with connections to matching-type problems which, among other graph problems, then are used to compute approximations for larger values of k.

Faisal Abu-Khzam, Cristina Bazgan, Katrin Casel, and Henning Fernau. Building Clusters with Lower-Bounded Sizes. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 4:1-4:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{abukhzam_et_al:LIPIcs.ISAAC.2016.4, author = {Abu-Khzam, Faisal and Bazgan, Cristina and Casel, Katrin and Fernau, Henning}, title = {{Building Clusters with Lower-Bounded Sizes}}, booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)}, pages = {4:1--4:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-026-2}, ISSN = {1868-8969}, year = {2016}, volume = {64}, editor = {Hong, Seok-Hee}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.4}, URN = {urn:nbn:de:0030-drops-67742}, doi = {10.4230/LIPIcs.ISAAC.2016.4}, annote = {Keywords: Clustering, Approximation Algorithms, Complexity, Matching} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

It is shown that the shortest-grammar problem remains NP-complete if the alphabet is fixed and has a size of at least 24 (which settles an open question). On the other hand, this problem can be solved in polynomial-time, if the number of nonterminals is bounded, which is shown by encoding the problem as a problem on graphs with interval structure. Furthermore, we present an O(3n) exact exponential-time algorithm, based on dynamic programming. Similar results are also given for 1-level grammars, i.e., grammars for which only the start rule contains nonterminals on the right side (thus, investigating the impact of the "hierarchical depth" on the complexity of the shortest-grammar problem).

Katrin Casel, Henning Fernau, Serge Gaspers, Benjamin Gras, and Markus L. Schmid. On the Complexity of Grammar-Based Compression over Fixed Alphabets. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 122:1-122:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{casel_et_al:LIPIcs.ICALP.2016.122, author = {Casel, Katrin and Fernau, Henning and Gaspers, Serge and Gras, Benjamin and Schmid, Markus L.}, title = {{On the Complexity of Grammar-Based Compression over Fixed Alphabets}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {122:1--122:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.122}, URN = {urn:nbn:de:0030-drops-62570}, doi = {10.4230/LIPIcs.ICALP.2016.122}, annote = {Keywords: Grammar-Based Compression, Straight-Line Programs, NP-Completeness, Exact Exponential Time Algorithms} }