License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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DOI: 10.4230/LIPIcs.MFCS.2020.35
URN: urn:nbn:de:0030-drops-127026
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Fomin, Fedor V. ; Sagunov, Danil ; Simonov, Kirill

Building Large k-Cores from Sparse Graphs

LIPIcs-MFCS-2020-35.pdf (0.5 MB)


A popular model to measure network stability is the k-core, that is the maximal induced subgraph in which every vertex has degree at least k. For example, k-cores are commonly used to model the unraveling phenomena in social networks. In this model, users having less than k connections within the network leave it, so the remaining users form exactly the k-core. In this paper we study the question of whether it is possible to make the network more robust by spending only a limited amount of resources on new connections. A mathematical model for the k-core construction problem is the following Edge k-Core optimization problem. We are given a graph G and integers k, b and p. The task is to ensure that the k-core of G has at least p vertices by adding at most b edges.
The previous studies on Edge k-Core demonstrate that the problem is computationally challenging. In particular, it is NP-hard when k = 3, W[1]-hard when parameterized by k+b+p (Chitnis and Talmon, 2018), and APX-hard (Zhou et al, 2019). Nevertheless, we show that there are efficient algorithms with provable guarantee when the k-core has to be constructed from a sparse graph with some additional structural properties. Our results are
- When the input graph is a forest, Edge k-Core is solvable in polynomial time;
- Edge k-Core is fixed-parameter tractable (FPT) when parameterized by the minimum size of a vertex cover in the input graph. On the other hand, with such parameterization, the problem does not admit a polynomial kernel subject to a widely-believed assumption from complexity theory;
- Edge k-Core is FPT parameterized by the treewidth of the graph plus k. This improves upon a result of Chitnis and Talmon by not requiring b to be small. Each of our algorithms is built upon a new graph-theoretical result interesting in its own.

BibTeX - Entry

  author =	{Fedor V. Fomin and Danil Sagunov and Kirill Simonov},
  title =	{{Building Large k-Cores from Sparse Graphs}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{35:1--35:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Javier Esparza and Daniel Kr{\'a}ΔΎ},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-127026},
  doi =		{10.4230/LIPIcs.MFCS.2020.35},
  annote =	{Keywords: parameterized complexity, k-core, vertex cover, treewidth}

Keywords: parameterized complexity, k-core, vertex cover, treewidth
Collection: 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)
Issue Date: 2020
Date of publication: 18.08.2020

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