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Complete Volume

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

LIPIcs, Volume 265, SEA 2023, Complete Volume

21st International Symposium on Experimental Algorithms (SEA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 265, pp. 1-390, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Proceedings{georgiadis:LIPIcs.SEA.2023, title = {{LIPIcs, Volume 265, SEA 2023, Complete Volume}}, booktitle = {21st International Symposium on Experimental Algorithms (SEA 2023)}, pages = {1--390}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2023}, URN = {urn:nbn:de:0030-drops-183495}, doi = {10.4230/LIPIcs.SEA.2023}, annote = {Keywords: LIPIcs, Volume 265, SEA 2023, Complete Volume} }

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Front Matter

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

Front Matter, Table of Contents, Preface, Conference Organization

21st International Symposium on Experimental Algorithms (SEA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 265, pp. 0:i-0:xii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{georgiadis:LIPIcs.SEA.2023.0, author = {Georgiadis, Loukas}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {21st International Symposium on Experimental Algorithms (SEA 2023)}, pages = {0:i--0:xii}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2023.0}, URN = {urn:nbn:de:0030-drops-183508}, doi = {10.4230/LIPIcs.SEA.2023.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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

The notions of edge-cuts and k-edge-connected components are fundamental in graph theory with numerous practical applications. Very recently, the first linear-time algorithms for computing all the 3-edge cuts and the 4-edge-connected components of a graph have been introduced. In this paper we present carefully engineered implementations of these algorithms and evaluate their efficiency in practice, by performing a thorough empirical study using both real-world graphs taken from a variety of application areas, as well as artificial graphs. To the best of our knowledge, this is the first experimental study for these problems, which highlights the merits and weaknesses of each technique. Furthermore, we present an improved algorithm for computing the 4-edge-connected components of an undirected graph in linear time. The new algorithm uses only elementary data structures, and is implementable in the pointer machine model of computation.

Loukas Georgiadis, Giuseppe F. Italiano, and Evangelos Kosinas. Computing the 4-Edge-Connected Components of a Graph: An Experimental Study. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 60:1-60:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{georgiadis_et_al:LIPIcs.ESA.2022.60, author = {Georgiadis, Loukas and Italiano, Giuseppe F. and Kosinas, Evangelos}, title = {{Computing the 4-Edge-Connected Components of a Graph: An Experimental Study}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {60:1--60:16}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.60}, URN = {urn:nbn:de:0030-drops-169988}, doi = {10.4230/LIPIcs.ESA.2022.60}, annote = {Keywords: Connectivity Cuts, Edge Connectivity, Graph Algorithms} }

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**Published in:** LIPIcs, Volume 233, 20th International Symposium on Experimental Algorithms (SEA 2022)

A classic result of Edmonds states that the maximum number of edge-disjoint arborescences of a directed graph G, rooted at a designated vertex s, equals the minimum cardinality c_G(s) of an s-cut of G. This concept is related to the edge connectivity λ(G) of a strongly connected directed graph G, defined as the minimum number of edges whose deletion leaves a graph that is not strongly connected. In this paper, we address the question of how efficiently we can compute a maximum packing of edge-disjoint arborescences in practice, compared to the time required to determine the edge connectivity of a graph. To that end, we explore the design space of efficient algorithms for packing arborescences of a directed graph in practice and conduct a thorough empirical study to highlight the merits and weaknesses of each technique. In particular, we present an efficient implementation of Gabow’s arborescence packing algorithm and provide a simple but efficient heuristic that significantly improves its running time in practice.

Loukas Georgiadis, Dionysios Kefallinos, Anna Mpanti, and Stavros D. Nikolopoulos. An Experimental Study of Algorithms for Packing Arborescences. In 20th International Symposium on Experimental Algorithms (SEA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 233, pp. 14:1-14:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{georgiadis_et_al:LIPIcs.SEA.2022.14, author = {Georgiadis, Loukas and Kefallinos, Dionysios and Mpanti, Anna and Nikolopoulos, Stavros D.}, title = {{An Experimental Study of Algorithms for Packing Arborescences}}, booktitle = {20th International Symposium on Experimental Algorithms (SEA 2022)}, pages = {14:1--14:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-251-8}, ISSN = {1868-8969}, year = {2022}, volume = {233}, editor = {Schulz, Christian and U\c{c}ar, Bora}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2022.14}, URN = {urn:nbn:de:0030-drops-165480}, doi = {10.4230/LIPIcs.SEA.2022.14}, annote = {Keywords: Arborescences, Edge Connectivity, Graph Algorithms} }

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

We present the first linear-time algorithm that computes the 4-edge-connected components of an undirected graph. Hence, we also obtain the first linear-time algorithm for testing 4-edge connectivity. Our results are based on a linear-time algorithm that computes the 3-edge cuts of a 3-edge-connected graph G, and a linear-time procedure that, given the collection of all 3-edge cuts, partitions the vertices of G into the 4-edge-connected components.

Loukas Georgiadis, Giuseppe F. Italiano, and Evangelos Kosinas. Computing the 4-Edge-Connected Components of a Graph in Linear Time. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 47:1-47:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{georgiadis_et_al:LIPIcs.ESA.2021.47, author = {Georgiadis, Loukas and Italiano, Giuseppe F. and Kosinas, Evangelos}, title = {{Computing the 4-Edge-Connected Components of a Graph in Linear Time}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {47:1--47:17}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.47}, URN = {urn:nbn:de:0030-drops-146286}, doi = {10.4230/LIPIcs.ESA.2021.47}, annote = {Keywords: Cuts, Edge Connectivity, Graph Algorithms} }

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**Published in:** LIPIcs, Volume 190, 19th International Symposium on Experimental Algorithms (SEA 2021)

We consider two problems regarding the computation of connectivity cuts in undirected graphs, namely identifying vertex-edge cut-pairs and identifying 2-edge cuts, and present an experimental study of efficient algorithms for their computation. In the first problem, we are given a biconnected graph G and our goal is to find all vertices v such that G⧵v is not 2-edge-connected, while in the second problem, we are given a 2-edge-connected graph G and our goal is to find all edges e such that G⧵e is not 2-edge-connected. These problems are motivated by the notion of twinless strong connectivity in directed graphs but are also of independent interest. Moreover, the computation of 2-edge cuts is a main step in algorithms that compute the 3-edge-connected components of a graph. In this paper, we present streamlined versions of two recent linear-time algorithms of Georgiadis and Kosinas that compute all vertex-edge cut-pairs and all 2-edge cuts, respectively. We compare the empirical performance of our vertex-edge cut-pairs algorithm with an alternative linear-time method that exploits the structure of the triconnected components of G. Also, we compare the empirical performance of our 2-edge cuts algorithm with the algorithm of Tsin, which was reported to be the fastest one among the previously existing for this problem. To that end, we conduct a thorough experimental study to highlight the merits and weaknesses of each technique.

Loukas Georgiadis, Konstantinos Giannis, Giuseppe F. Italiano, and Evangelos Kosinas. Computing Vertex-Edge Cut-Pairs and 2-Edge Cuts in Practice. In 19th International Symposium on Experimental Algorithms (SEA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 190, pp. 20:1-20:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{georgiadis_et_al:LIPIcs.SEA.2021.20, author = {Georgiadis, Loukas and Giannis, Konstantinos and Italiano, Giuseppe F. and Kosinas, Evangelos}, title = {{Computing Vertex-Edge Cut-Pairs and 2-Edge Cuts in Practice}}, booktitle = {19th International Symposium on Experimental Algorithms (SEA 2021)}, pages = {20:1--20:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-185-6}, ISSN = {1868-8969}, year = {2021}, volume = {190}, editor = {Coudert, David and Natale, Emanuele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2021.20}, URN = {urn:nbn:de:0030-drops-137920}, doi = {10.4230/LIPIcs.SEA.2021.20}, annote = {Keywords: 2-Connectivity, Graph Algorithms, Split Components} }

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**Published in:** LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

A directed graph G = (V,E) is twinless strongly connected if it contains a strongly connected spanning subgraph without any pair of antiparallel (or twin) edges. The twinless strongly connected components (TSCCs) of a directed graph G are its maximal twinless strongly connected subgraphs. These concepts have several diverse applications, such as the design of telecommunication networks and the structural stability of buildings. A vertex v ∈ V is a twinless strong articulation point of G, if the deletion of v increases the number of TSCCs of G. Here, we present the first linear-time algorithm that finds all the twinless strong articulation points of a directed graph. We show that the computation of twinless strong articulation points reduces to the following problem in undirected graphs, which may be of independent interest: Given a 2-vertex-connected undirected graph H, find all vertices v for which there exists an edge e such that H⧵{v,e} is not connected. We develop a linear-time algorithm that not only finds all such vertices v, but also computes the number of edges e such that H⧵{v,e} is not connected. This also implies that for each twinless strong articulation point v which is not a strong articulation point in a strongly connected digraph G, we can compute the number of TSCCs in G⧵v.

Loukas Georgiadis and Evangelos Kosinas. Linear-Time Algorithms for Computing Twinless Strong Articulation Points and Related Problems. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{georgiadis_et_al:LIPIcs.ISAAC.2020.38, author = {Georgiadis, Loukas and Kosinas, Evangelos}, title = {{Linear-Time Algorithms for Computing Twinless Strong Articulation Points and Related Problems}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {38:1--38:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-173-3}, ISSN = {1868-8969}, year = {2020}, volume = {181}, editor = {Cao, Yixin and Cheng, Siu-Wing and Li, Minming}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.38}, URN = {urn:nbn:de:0030-drops-133820}, doi = {10.4230/LIPIcs.ISAAC.2020.38}, annote = {Keywords: 2-connectivity, cut pairs, strongly connected components} }

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**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

We consider practical algorithms for maintaining the dominator tree and a low-high order in directed acyclic graphs (DAGs) subject to dynamic operations. Let G be a directed graph with a distinguished start vertex s. The dominator tree D of G is a tree rooted at s, such that a vertex v is an ancestor of a vertex w if and only if all paths from s to w in G include v. The dominator tree is a central tool in program optimization and code generation, and has many applications in other diverse areas including constraint programming, circuit testing, biology, and in algorithms for graph connectivity problems. A low-high order of G is a preorder of D that certifies the correctness of D, and has further applications in connectivity and path-determination problems.
We first provide a practical and carefully engineered version of a recent algorithm [ICALP 2017] for maintaining the dominator tree of a DAG through a sequence of edge deletions. The algorithm runs in O(mn) total time and O(m) space, where n is the number of vertices and m is the number of edges before any deletion. In addition, we present a new algorithm that maintains a low-high order of a DAG under edge deletions within the same bounds. Both results extend to the case of reducible graphs (a class that includes DAGs). Furthermore, we present a fully dynamic algorithm for maintaining the dominator tree of a DAG under an intermixed sequence of edge insertions and deletions. Although it does not maintain the O(mn) worst-case bound of the decremental algorithm, our experiments highlight that the fully dynamic algorithm performs very well in practice. Finally, we study the practical efficiency of all our algorithms by conducting an extensive experimental study on real-world and synthetic graphs.

Loukas Georgiadis, Konstantinos Giannis, Giuseppe F. Italiano, Aikaterini Karanasiou, and Luigi Laura. Dynamic Dominators and Low-High Orders in DAGs. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 50:1-50:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{georgiadis_et_al:LIPIcs.ESA.2019.50, author = {Georgiadis, Loukas and Giannis, Konstantinos and Italiano, Giuseppe F. and Karanasiou, Aikaterini and Laura, Luigi}, title = {{Dynamic Dominators and Low-High Orders in DAGs}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {50:1--50:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.50}, URN = {urn:nbn:de:0030-drops-111715}, doi = {10.4230/LIPIcs.ESA.2019.50}, annote = {Keywords: Connectivity, dominators, low-high orders} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

The All-Pairs Min-Cut problem (aka All-Pairs Max-Flow) asks to compute a minimum s-t cut (or just its value) for all pairs of vertices s,t. We study this problem in directed graphs with unit edge/vertex capacities (corresponding to edge/vertex connectivity). Our focus is on the k-bounded case, where the algorithm has to find all pairs with min-cut value less than k, and report only those. The most basic case k=1 is the Transitive Closure (TC) problem, which can be solved in graphs with n vertices and m edges in time O(mn) combinatorially, and in time O(n^{omega}) where omega<2.38 is the matrix-multiplication exponent. These time bounds are conjectured to be optimal.
We present new algorithms and conditional lower bounds that advance the frontier for larger k, as follows:
- A randomized algorithm for vertex capacities that runs in time {O}((nk)^{omega}). This is only a factor k^omega away from the TC bound, and nearly matches it for all k=n^{o(1)}.
- Two deterministic algorithms for edge capacities (which is more general) that work in DAGs and further reports a minimum cut for each pair. The first algorithm is combinatorial (does not involve matrix multiplication) and runs in time {O}(2^{{O}(k^2)}* mn). The second algorithm can be faster on dense DAGs and runs in time {O}((k log n)^{4^{k+o(k)}}* n^{omega}). Previously, Georgiadis et al. [ICALP 2017], could match the TC bound (up to n^{o(1)} factors) only when k=2, and now our two algorithms match it for all k=o(sqrt{log n}) and k=o(log log n).
- The first super-cubic lower bound of n^{omega-1-o(1)} k^2 time under the 4-Clique conjecture, which holds even in the simplest case of DAGs with unit vertex capacities. It improves on the previous (SETH-based) lower bounds even in the unbounded setting k=n. For combinatorial algorithms, our reduction implies an n^{2-o(1)} k^2 conditional lower bound. Thus, we identify new settings where the complexity of the problem is (conditionally) higher than that of TC.
Our three sets of results are obtained via different techniques. The first one adapts the network coding method of Cheung, Lau, and Leung [SICOMP 2013] to vertex-capacitated digraphs. The second set exploits new insights on the structure of latest cuts together with suitable algebraic tools. The lower bounds arise from a novel reduction of a different structure than the SETH-based constructions.

Amir Abboud, Loukas Georgiadis, Giuseppe F. Italiano, Robert Krauthgamer, Nikos Parotsidis, Ohad Trabelsi, Przemysław Uznański, and Daniel Wolleb-Graf. Faster Algorithms for All-Pairs Bounded Min-Cuts. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{abboud_et_al:LIPIcs.ICALP.2019.7, author = {Abboud, Amir and Georgiadis, Loukas and Italiano, Giuseppe F. and Krauthgamer, Robert and Parotsidis, Nikos and Trabelsi, Ohad and Uzna\'{n}ski, Przemys{\l}aw and Wolleb-Graf, Daniel}, title = {{Faster Algorithms for All-Pairs Bounded Min-Cuts}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {7:1--7:15}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.7}, URN = {urn:nbn:de:0030-drops-105833}, doi = {10.4230/LIPIcs.ICALP.2019.7}, annote = {Keywords: All-pairs min-cut, k-reachability, network coding, Directed graphs, fine-grained complexity} }

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**Published in:** LIPIcs, Volume 75, 16th International Symposium on Experimental Algorithms (SEA 2017)

Let G = (V, E) be a 2-vertex-connected directed graph with m edges and n vertices. We consider the problem of approximating the smallest 2-vertex connected spanning subgraph (2VCSS) of G, and provide new efficient algorithms for this problem based on a clever use of low-high orders. The best previously known algorithms were able to compute a 3/2-approximation in O(m n+n 2) time, or a 3-approximation faster in linear time. In this paper, we present a linear-time algorithm that achieves a better approximation ratio of 2, and another algorithm that matches the previous 3/2-approximation in O(m n + n 2 ) time. We conducted a thorough experimental evaluation of all the above algorithms on a variety of input graphs. The experimental results show that both our two new algorithms perform well in practice. In particular, in our experiments the new 3/2-approximation algorithm was always faster than the previous 3/2-approximation algorithm, while their two approximation ratios were close. On the other side, our new linear-time algorithm yielded consistently better approximation ratios than the previously known linear-time algorithm, at the price of a small overhead in the running time.

Loukas Georgiadis, Giuseppe F. Italiano, and Aikaterini Karanasiou. Approximating the Smallest 2-Vertex-Connected Spanning Subgraph via Low-High Orders. In 16th International Symposium on Experimental Algorithms (SEA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 75, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{georgiadis_et_al:LIPIcs.SEA.2017.9, author = {Georgiadis, Loukas and Italiano, Giuseppe F. and Karanasiou, Aikaterini}, title = {{Approximating the Smallest 2-Vertex-Connected Spanning Subgraph via Low-High Orders}}, booktitle = {16th International Symposium on Experimental Algorithms (SEA 2017)}, pages = {9:1--9:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-036-1}, ISSN = {1868-8969}, year = {2017}, volume = {75}, editor = {Iliopoulos, Costas S. and Pissis, Solon P. and Puglisi, Simon J. and Raman, Rajeev}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2017.9}, URN = {urn:nbn:de:0030-drops-76299}, doi = {10.4230/LIPIcs.SEA.2017.9}, annote = {Keywords: 2-vertex connectivity, approximation algorithms, directed graphs} }

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**Published in:** LIPIcs, Volume 75, 16th International Symposium on Experimental Algorithms (SEA 2017)

A flow graph G = (V, E, s) is a directed graph with a distinguished start vertex s. The dominator tree D of G is a tree rooted at s, such that a vertex v is an ancestor of a vertex w if and only if all paths from s to w include v. The dominator tree is a central tool in program optimization and code generation, and has many applications in other diverse areas including constraint programming, circuit testing, biology, and in algorithms for graph connectivity problems. A low-high order of G is a preorder d of D that certifies the correctness of D, and has further applications in connectivity and path-determination problems.
In this paper we consider how to maintain efficiently a low-high order of a flow graph incrementally under edge insertions. We present algorithms that run in O(mn) total time for a sequence of edge insertions in a flow graph with n vertices, where m is the total number of edges after all insertions. These immediately provide the first incremental certifying algorithms for maintaining the dominator tree in O(mn) total time, and also imply incremental algorithms for other problems. Hence, we provide a substantial improvement over the O(m^2) straightforward algorithms, which recompute the solution from scratch after each edge insertion. Furthermore, we provide efficient implementations of our algorithms and conduct an extensive experimental study on real-world graphs taken from a variety of application areas. The experimental results show that our algorithms perform very well in practice.

Loukas Georgiadis, Konstantinos Giannis, Aikaterini Karanasiou, and Luigi Laura. Incremental Low-High Orders of Directed Graphs and Applications. In 16th International Symposium on Experimental Algorithms (SEA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 75, pp. 27:1-27:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{georgiadis_et_al:LIPIcs.SEA.2017.27, author = {Georgiadis, Loukas and Giannis, Konstantinos and Karanasiou, Aikaterini and Laura, Luigi}, title = {{Incremental Low-High Orders of Directed Graphs and Applications}}, booktitle = {16th International Symposium on Experimental Algorithms (SEA 2017)}, pages = {27:1--27:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-036-1}, ISSN = {1868-8969}, year = {2017}, volume = {75}, editor = {Iliopoulos, Costas S. and Pissis, Solon P. and Puglisi, Simon J. and Raman, Rajeev}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2017.27}, URN = {urn:nbn:de:0030-drops-76319}, doi = {10.4230/LIPIcs.SEA.2017.27}, annote = {Keywords: connectivity, directed graphs, dominators, dynamic algorithms} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We introduce a new dynamic data structure for maintaining the strongly connected components (SCCs) of a directed graph (digraph) under edge deletions, so as to answer a rich repertoire of connectivity queries. Our main technical contribution is a decremental data structure that supports sensitivity queries of the form "are u and v strongly connected in the graph G \ w?", for any triple of vertices u, v, w, while G undergoes deletions of edges. Our data structure processes a sequence of edge deletions in a digraph with $n$ vertices in O(m n log n) total time and O(n^2 log n) space, where m is the number of edges before any deletion, and answers the above queries in constant time. We can leverage our data structure to obtain decremental data structures for many more types of queries within the same time and space complexity. For instance for edge-related queries, such as testing whether two query vertices u and v are strongly connected in G \ e, for some query edge e.
As another important application of our decremental data structure, we provide the first nontrivial algorithm for maintaining the dominator tree of a flow graph under edge deletions. We present an algorithm that processes a sequence of edge deletions in a flow graph in O(m n log n) total time and O(n^2 log n) space. For reducible flow graphs we provide an O(mn)-time and O(m + n)-space algorithm. We give a conditional lower bound that provides evidence that these running times may be tight up to subpolynomial factors.

Loukas Georgiadis, Thomas Dueholm Hansen, Giuseppe F. Italiano, Sebastian Krinninger, and Nikos Parotsidis. Decremental Data Structures for Connectivity and Dominators in Directed Graphs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 42:1-42:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{georgiadis_et_al:LIPIcs.ICALP.2017.42, author = {Georgiadis, Loukas and Dueholm Hansen, Thomas and Italiano, Giuseppe F. and Krinninger, Sebastian and Parotsidis, Nikos}, title = {{Decremental Data Structures for Connectivity and Dominators in Directed Graphs}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {42:1--42:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.42}, URN = {urn:nbn:de:0030-drops-74455}, doi = {10.4230/LIPIcs.ICALP.2017.42}, annote = {Keywords: dynamic graph algorithms, decremental algorithms, dominator tree, strong connectivity under failures} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

In the 2-reachability problem we are given a directed graph G and we wish to determine if there are two (edge or vertex) disjoint paths from u to v, for given pair of vertices u and v. In this paper, we present an algorithm that computes 2-reachability information for all pairs of vertices in O(n^w log n) time, where n is the number of vertices and w is the matrix multiplication exponent. Hence, we show that the running time of all-pairs 2-reachability is only within a log factor of transitive closure. Moreover, our algorithm produces a witness (i.e., a separating edge or a separating vertex) for all pair of vertices where 2-reachability does not hold. By processing these witnesses, we can compute all the edge- and vertex-dominator trees of G in O(n^2) additional time, which in turn enables us to answer various connectivity queries in O(1) time. For instance, we can test in constant time if there is a path from u to v avoiding an edge e, for any pair of query vertices u and v, and any query edge e, or if there is a path from u to v avoiding a vertex w, for any query vertices u, v, and w.

Loukas Georgiadis, Daniel Graf, Giuseppe F. Italiano, Nikos Parotsidis, and Przemyslaw Uznanski. All-Pairs 2-Reachability in O(n^w log n) Time. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 74:1-74:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{georgiadis_et_al:LIPIcs.ICALP.2017.74, author = {Georgiadis, Loukas and Graf, Daniel and Italiano, Giuseppe F. and Parotsidis, Nikos and Uznanski, Przemyslaw}, title = {{All-Pairs 2-Reachability in O(n^w log n) Time}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {74:1--74:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.74}, URN = {urn:nbn:de:0030-drops-74510}, doi = {10.4230/LIPIcs.ICALP.2017.74}, annote = {Keywords: 2-reachability, All Dominator Trees, Directed Graphs, Boolean Matrix Multiplication} }

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

We present an algorithm that can update the 2-edge-connected blocks of a directed graph with n vertices through a sequence of m edge insertions in a total of O(m*n) time. After each insertion, we can answer the following queries in asymptotically optimal time:
- Test in constant time if two query vertices v and w are 2-edge-connected. Moreover, if v and w are not 2-edge-connected, we can produce in constant time a “witness” of this property, by exhibiting an edge that is contained in all paths from v to w or in all paths from w to v.
- Report in O(n) time all the 2-edge-connected blocks of G.
This is the first dynamic algorithm for 2-connectivity problems on directed graphs, and it matches the best known bounds for simpler problems, such as incremental transitive closure.

Loukas Georgiadis, Giuseppe F. Italiano, and Nikos Parotsidis. Incremental 2-Edge-Connectivity in Directed Graphs. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 49:1-49:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{georgiadis_et_al:LIPIcs.ICALP.2016.49, author = {Georgiadis, Loukas and Italiano, Giuseppe F. and Parotsidis, Nikos}, title = {{Incremental 2-Edge-Connectivity in Directed Graphs}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {49:1--49:15}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.49}, URN = {urn:nbn:de:0030-drops-63292}, doi = {10.4230/LIPIcs.ICALP.2016.49}, annote = {Keywords: 2-edge connectivity on directed graphs; dynamic graph algorithms; incremental algorithms.} }

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Invited Talk

**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

We survey some recent results on 2-edge and 2-vertex connectivity problems in directed graphs. Despite being complete analogs of the corresponding notions on undirected graphs, in digraphs 2-vertex and 2-edge connectivity have a much richer and more complicated structure. It is thus not surprising that 2-connectivity problems on directed graphs appear to be more difficult than on undirected graphs. For undirected graphs it has been known for over 40 years how to compute all bridges, articulation points, 2-edge- and 2-vertex-connected components in linear time, by simply using depth-first search. In the case of digraphs, however, the very same problems have been much more challenging and required the development of new tools and techniques.

Loukas Georgiadis, Giuseppe F. Italiano, and Nikos Parotsidis. 2-Connectivity in Directed Graphs (Invited Talk). In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 1:1-1:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{georgiadis_et_al:LIPIcs.ESA.2016.1, author = {Georgiadis, Loukas and Italiano, Giuseppe F. and Parotsidis, Nikos}, title = {{2-Connectivity in Directed Graphs}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {1:1--1:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.1}, URN = {urn:nbn:de:0030-drops-63451}, doi = {10.4230/LIPIcs.ESA.2016.1}, annote = {Keywords: 2-edge and 2-vertex connectivity on directed graphs, graph algorithms, dominator trees} }

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