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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

In the Vertex Connectivity Survivable Network Design (VC-SNDP) problem, the input is a graph G and a function d: V(G) × V(G) → ℕ that encodes the vertex-connectivity demands between pairs of vertices. The objective is to find the smallest subgraph H of G that satisfies all these demands. It is a well-studied NP-complete problem that generalizes several network design problems. We consider the case of uniform demands, where for every vertex pair (u,v) the connectivity demand d(u,v) is a fixed integer κ. It is an important problem with wide applications.
We study this problem in the realm of Parameterized Complexity. In this setting, in addition to G and d we are given an integer 𝓁 as the parameter and the objective is to determine if we can remove at least 𝓁 edges from G without violating any connectivity constraints. This was posed as an open problem by Bang-Jansen et.al. [SODA 2018], who studied the edge-connectivity variant of the problem under the same settings. Using a powerful classification result of Lokshtanov et al. [ICALP 2018], Gutin et al. [JCSS 2019] recently showed that this problem admits a (non-uniform) FPT algorithm where the running time was unspecified. Further they also gave an (uniform) FPT algorithm for the case of κ = 2. In this paper we present a (uniform) FPT algorithm any κ that runs in time 2^{O(κ² 𝓁⁴ log 𝓁)}⋅ |V(G)|^O(1).
Our algorithm is built upon new insights on vertex connectivity in graphs. Our main conceptual contribution is a novel graph decomposition called the Wheel decomposition. Informally, it is a partition of the edge set of a graph G, E(G) = X₁ ∪ X₂ … ∪ X_r, with the parts arranged in a cyclic order, such that each vertex v ∈ V(G) either has edges in at most two consecutive parts, or has edges in every part of this partition. The first kind of vertices can be thought of as the rim of the wheel, while the second kind form the hub. Additionally, the vertex cuts induced by these edge-sets in G have highly symmetric properties. Our main technical result, informally speaking, establishes that "nearly edge-minimal’’ κ-vertex connected graphs admit a wheel decomposition - a fact that can be exploited for designing algorithms. We believe that this decomposition is of independent interest and it could be a useful tool in resolving other open problems.

Jørgen Bang-Jensen, Kristine Vitting Klinkby, Pranabendu Misra, and Saket Saurabh. A Parameterized Algorithm for Vertex Connectivity Survivable Network Design Problem with Uniform Demands. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 13:1-13:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bangjensen_et_al:LIPIcs.ESA.2023.13, author = {Bang-Jensen, J{\o}rgen and Klinkby, Kristine Vitting and Misra, Pranabendu and Saurabh, Saket}, title = {{A Parameterized Algorithm for Vertex Connectivity Survivable Network Design Problem with Uniform Demands}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {13:1--13:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.13}, URN = {urn:nbn:de:0030-drops-186663}, doi = {10.4230/LIPIcs.ESA.2023.13}, annote = {Keywords: Parameterized Complexity, Vertex Connectivity, Network Design} }

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

Unlike the problem of deciding whether a digraph D = (V,A) has 𝓁 in-branchings (or 𝓁 out-branchings) is polynomial time solvable, the problem of deciding whether a digraph D = (V,A) has an in-branching B^- and an out-branching B^+ which are arc-disjoint is NP-complete. Motivated by this, a natural optimization question that has been studied in the realm of Parameterized Complexity is called Rooted k-Distinct Branchings. In this problem, a digraph D = (V,A) with two prescribed vertices s,t are given as input and the question is whether D has an in-branching rooted at t and an out-branching rooted at s such that they differ on at least k arcs. Bang-Jensen et al. [Algorithmica, 2016 ] showed that the problem is fixed parameter tractable (FPT) on strongly connected digraphs. Gutin et al. [ICALP, 2017; JCSS, 2018 ] completely resolved this problem by designing an algorithm with running time 2^{𝒪(k² log² k)}n^{𝒪(1)}. Here, n denotes the number of vertices of the input digraph. In this paper, answering an open question of Gutin et al., we design a polynomial kernel for Rooted k-Distinct Branchings. In particular, we obtain the following: Given an instance (D,k,s,t) of Rooted k-Distinct Branchings, in polynomial time we obtain an equivalent instance (D',k',s,t) of Rooted k-Distinct Branchings such that |V(D')| ≤ 𝒪(k²) and the treewidth of the underlying undirected graph is at most 𝒪(k). This result immediately yields an FPT algorithm with running time 2^{𝒪(klog k)}+ n^{𝒪(1)}; improving upon the previous running time of Gutin et al. For our algorithms, we prove a structural result about paths avoiding many arcs in a given in-branching or out-branching. This result might turn out to be useful for getting other results for problems concerning in-and out-branchings.

Jørgen Bang-Jensen, Kristine Vitting Klinkby, and Saket Saurabh. k-Distinct Branchings Admits a Polynomial Kernel. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 11:1-11:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bangjensen_et_al:LIPIcs.ESA.2021.11, author = {Bang-Jensen, J{\o}rgen and Klinkby, Kristine Vitting and Saurabh, Saket}, title = {{k-Distinct Branchings Admits a Polynomial Kernel}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {11:1--11:15}, 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.2021.11}, URN = {urn:nbn:de:0030-drops-145925}, doi = {10.4230/LIPIcs.ESA.2021.11}, annote = {Keywords: Digraphs, Polynomial Kernel, In-branching, Out-Branching} }

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