5 Search Results for "Bang-Jensen, Jørgen"

Document
DOI: 10.4230/LIPIcs.ESA.2023.13
A Parameterized Algorithm for Vertex Connectivity Survivable Network Design Problem with Uniform Demands

Authors: Jørgen Bang-Jensen, Kristine Vitting Klinkby, Pranabendu Misra, and Saket Saurabh

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

Abstract
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.
Cite as

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-dev.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}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2021.8
A Succinct Formalization of the Completeness of First-Order Logic

Authors: Asta Halkjær From

Published in: LIPIcs, Volume 239, 27th International Conference on Types for Proofs and Programs (TYPES 2021)

Abstract
I succinctly formalize the soundness and completeness of a small Hilbert system for first-order logic in the proof assistant Isabelle/HOL. The proof combines and details ideas from de Bruijn, Henkin, Herbrand, Hilbert, Hintikka, Lindenbaum, Smullyan and others in a novel way, and I use a declarative style, custom notation and proof automation to obtain a readable formalization. The formalized definitions of Hintikka sets and Herbrand structures allow open and closed formulas to be treated uniformly, making free variables a non-concern. This paper collects important techniques in mathematical logic in a way suited for both study and further work.
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Asta Halkjær From. A Succinct Formalization of the Completeness of First-Order Logic. In 27th International Conference on Types for Proofs and Programs (TYPES 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 239, pp. 8:1-8:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{from:LIPIcs.TYPES.2021.8,
  author =	{From, Asta Halkj{\ae}r},
  title =	{{A Succinct Formalization of the Completeness of First-Order Logic}},
  booktitle =	{27th International Conference on Types for Proofs and Programs (TYPES 2021)},
  pages =	{8:1--8:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-254-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{239},
  editor =	{Basold, Henning and Cockx, Jesper and Ghilezan, Silvia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2021.8},
  URN =		{urn:nbn:de:0030-drops-167771},
  doi =		{10.4230/LIPIcs.TYPES.2021.8},
  annote =	{Keywords: First-Order Logic, Completeness, Isabelle/HOL}
}
Document
DOI: 10.4230/LIPIcs.ITP.2022.13
Verifying a Sequent Calculus Prover for First-Order Logic with Functions in Isabelle/HOL

Authors: Asta Halkjær From and Frederik Krogsdal Jacobsen

Published in: LIPIcs, Volume 237, 13th International Conference on Interactive Theorem Proving (ITP 2022)

Abstract
We describe the design, implementation and verification of an automated theorem prover for first-order logic with functions. The proof search procedure is based on sequent calculus and we formally verify its soundness and completeness in Isabelle/HOL using an existing abstract framework for coinductive proof trees. Our analytic completeness proof covers both open and closed formulas. Since our deterministic prover considers only the subset of terms relevant to proving a given sequent, we do so as well when building a countermodel from a failed proof. Finally, we formally connect our prover with the proof system and semantics of the existing SeCaV system. In particular, the prover can generate human-readable SeCaV proofs which are also machine-verifiable proof certificates.
Cite as

Asta Halkjær From and Frederik Krogsdal Jacobsen. Verifying a Sequent Calculus Prover for First-Order Logic with Functions in Isabelle/HOL. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 13:1-13:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{from_et_al:LIPIcs.ITP.2022.13,
  author =	{From, Asta Halkj{\ae}r and Jacobsen, Frederik Krogsdal},
  title =	{{Verifying a Sequent Calculus Prover for First-Order Logic with Functions in Isabelle/HOL}},
  booktitle =	{13th International Conference on Interactive Theorem Proving (ITP 2022)},
  pages =	{13:1--13:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-252-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{237},
  editor =	{Andronick, June and de Moura, Leonardo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2022.13},
  URN =		{urn:nbn:de:0030-drops-167221},
  doi =		{10.4230/LIPIcs.ITP.2022.13},
  annote =	{Keywords: Isabelle/HOL, SeCaV, First-Order Logic, Prover, Soundness, Completeness}
}
Document
DOI: 10.4230/LIPIcs.ESA.2021.11
k-Distinct Branchings Admits a Polynomial Kernel

Authors: Jørgen Bang-Jensen, Kristine Vitting Klinkby, and Saket Saurabh

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

Abstract
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.
Cite as

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-dev.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}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2020.2
Component Order Connectivity in Directed Graphs

Authors: Jørgen Bang-Jensen, Eduard Eiben, Gregory Gutin, Magnus Wahlström, and Anders Yeo

Published in: LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)

Abstract
A directed graph D is semicomplete if for every pair x,y of vertices of D, there is at least one arc between x and y. Thus, a tournament is a semicomplete digraph. In the Directed Component Order Connectivity (DCOC) problem, given a digraph D = (V,A) and a pair of natural numbers k and 𝓁, we are to decide whether there is a subset X of V of size k such that the largest strong connectivity component in D-X has at most 𝓁 vertices. Note that DCOC reduces to the Directed Feedback Vertex Set problem for 𝓁 = 1. We study parameterized complexity of DCOC for general and semicomplete digraphs with the following parameters: k, 𝓁, 𝓁+k and n-𝓁. In particular, we prove that DCOC with parameter k on semicomplete digraphs can be solved in time O^*(2^(16k)) but not in time O^*(2^o(k)) unless the Exponential Time Hypothesis (ETH) fails. The upper bound O^*(2^(16k)) implies the upper bound O^*(2^(16(n-𝓁))) for the parameter n-𝓁. We complement the latter by showing that there is no algorithm of time complexity O^*(2^o(n-𝓁)) unless ETH fails. Finally, we improve (in dependency on 𝓁) the upper bound of Göke, Marx and Mnich (2019) for the time complexity of DCOC with parameter 𝓁+k on general digraphs from O^*(2^O(k𝓁 log (k𝓁))) to O^*(2^O(klog (k𝓁))). Note that Drange, Dregi and van 't Hof (2016) proved that even for the undirected version of DCOC on split graphs there is no algorithm of running time O^*(2^o(klog 𝓁)) unless ETH fails and it is a long-standing problem to decide whether Directed Feedback Vertex Set admits an algorithm of time complexity O^*(2^o(klog k)).
Cite as

Jørgen Bang-Jensen, Eduard Eiben, Gregory Gutin, Magnus Wahlström, and Anders Yeo. Component Order Connectivity in Directed Graphs. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 2:1-2:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bangjensen_et_al:LIPIcs.IPEC.2020.2,
  author =	{Bang-Jensen, J{\o}rgen and Eiben, Eduard and Gutin, Gregory and Wahlstr\"{o}m, Magnus and Yeo, Anders},
  title =	{{Component Order Connectivity in Directed Graphs}},
  booktitle =	{15th International Symposium on Parameterized and Exact Computation (IPEC 2020)},
  pages =	{2:1--2:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-172-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{180},
  editor =	{Cao, Yixin and Pilipczuk, Marcin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.2},
  URN =		{urn:nbn:de:0030-drops-133058},
  doi =		{10.4230/LIPIcs.IPEC.2020.2},
  annote =	{Keywords: Parameterized Algorithms, component order connectivity, directed graphs, semicomplete digraphs}
}
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