3 Search Results for "Niklanovits, Aikaterini"

Document
DOI: 10.4230/LIPIcs.IPEC.2024.1
Combining Crown Structures for Vulnerability Measures

Authors: Katrin Casel, Tobias Friedrich, Aikaterini Niklanovits, Kirill Simonov, and Ziena Zeif

Published in: LIPIcs, Volume 321, 19th International Symposium on Parameterized and Exact Computation (IPEC 2024)

Abstract
Over the past decades, various metrics have emerged in graph theory to grasp the complex nature of network vulnerability. In this paper, we study two specific measures: (weighted) vertex integrity (wVI) and (weighted) component order connectivity (wCOC). These measures not only evaluate the number of vertices that need to be removed to decompose a graph into fragments, but also take into account the size of the largest remaining component. The main focus of our paper is on kernelization algorithms tailored to both measures. We capitalize on the structural attributes inherent in different crown decompositions, strategically combining them to introduce novel kernelization algorithms that advance the current state of the field. In particular, we extend the scope of the balanced crown decomposition provided by Casel et al. [Katrin Casel et al., 2021] and expand the applicability of crown decomposition techniques. In summary, we improve the vertex kernel of VI from p³ to 3p², and of wVI from p³ to 3(p² + p^{1.5} p_𝓁), where p_𝓁 < p represents the weight of the heaviest component after removing a solution. For wCOC we improve the vertex kernel from 𝒪(k²W + kW²) to 3μ(k + √{μ}W), where μ = max(k,W). We also give a combinatorial algorithm that provides a 2kW vertex kernel in fixed-parameter tractable time when parameterized by r, where r ≤ k is the size of a maximum (W+1)-packing. We further show that the algorithm computing the 2kW vertex kernel for COC can be transformed into a polynomial algorithm for two special cases, namely when W = 1, which corresponds to the well-known vertex cover problem, and for claw-free graphs. In particular, we show a new way to obtain a 2k vertex kernel (or to obtain a 2-approximation) for the vertex cover problem by only using crown structures.
Cite as

Katrin Casel, Tobias Friedrich, Aikaterini Niklanovits, Kirill Simonov, and Ziena Zeif. Combining Crown Structures for Vulnerability Measures. In 19th International Symposium on Parameterized and Exact Computation (IPEC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 321, pp. 1:1-1:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{casel_et_al:LIPIcs.IPEC.2024.1,
  author =	{Casel, Katrin and Friedrich, Tobias and Niklanovits, Aikaterini and Simonov, Kirill and Zeif, Ziena},
  title =	{{Combining Crown Structures for Vulnerability Measures}},
  booktitle =	{19th International Symposium on Parameterized and Exact Computation (IPEC 2024)},
  pages =	{1:1--1:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-353-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{321},
  editor =	{Bonnet, \'{E}douard and Rz\k{a}\.{z}ewski, Pawe{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2024.1},
  URN =		{urn:nbn:de:0030-drops-222270},
  doi =		{10.4230/LIPIcs.IPEC.2024.1},
  annote =	{Keywords: Crown Decomposition, Kernelization, Vertex Integrity, Component Order Connectivity}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.27
Connected k-Partition of k-Connected Graphs and c-Claw-Free Graphs

Authors: Ralf Borndörfer, Katrin Casel, Davis Issac, Aikaterini Niklanovits, Stephan Schwartz, and Ziena Zeif

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.ESA.2021.26
Balanced Crown Decomposition for Connectivity Constraints

Authors: Katrin Casel, Tobias Friedrich, Davis Issac, Aikaterini Niklanovits, and Ziena Zeif

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

Abstract
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.
Cite as

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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