2 Search Results for "Karpov, Nikolay"

Document
DOI: 10.4230/LIPIcs.ESA.2019.47
Going Far From Degeneracy

Authors: Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi

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

Abstract
An undirected graph G is d-degenerate if every subgraph of G has a vertex of degree at most d. By the classical theorem of Erdős and Gallai from 1959, every graph of degeneracy d>1 contains a cycle of length at least d+1. The proof of Erdős and Gallai is constructive and can be turned into a polynomial time algorithm constructing a cycle of length at least d+1. But can we decide in polynomial time whether a graph contains a cycle of length at least d+2? An easy reduction from Hamiltonian Cycle provides a negative answer to this question: Deciding whether a graph has a cycle of length at least d+2 is NP-complete. Surprisingly, the complexity of the problem changes drastically when the input graph is 2-connected. In this case we prove that deciding whether G contains a cycle of length at least d+k can be done in time 2^{O(k)}|V(G)|^O(1). In other words, deciding whether a 2-connected n-vertex G contains a cycle of length at least d+log{n} can be done in polynomial time. Similar algorithmic results hold for long paths in graphs. We observe that deciding whether a graph has a path of length at least d+1 is NP-complete. However, we prove that if graph G is connected, then deciding whether G contains a path of length at least d+k can be done in time 2^{O(k)}n^O(1). We complement these results by showing that the choice of degeneracy as the "above guarantee parameterization" is optimal in the following sense: For any epsilon>0 it is NP-complete to decide whether a connected (2-connected) graph of degeneracy d has a path (cycle) of length at least (1+epsilon)d.
Cite as

Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Going Far From Degeneracy. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 47:1-47:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{fomin_et_al:LIPIcs.ESA.2019.47,
  author =	{Fomin, Fedor V. and Golovach, Petr A. and Lokshtanov, Daniel and Panolan, Fahad and Saurabh, Saket and Zehavi, Meirav},
  title =	{{Going Far From Degeneracy}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{47:1--47:14},
  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.2019.47},
  URN =		{urn:nbn:de:0030-drops-111688},
  doi =		{10.4230/LIPIcs.ESA.2019.47},
  annote =	{Keywords: Longest path, longest cycle, fixed-parameter tractability, above guarantee parameterization}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2015.408
Parameterized Complexity of Secluded Connectivity Problems

Authors: Fedor V. Fomin, Petr A. Golovach, Nikolay Karpov, and Alexander S. Kulikov

Published in: LIPIcs, Volume 45, 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)

Abstract
The Secluded Path problem introduced by Chechik et al. in [ESA 2013] models a situation where a sensitive information has to be transmitted between a pair of nodes along a path in a network. The measure of the quality of a selected path is its exposure, which is the total weight of vertices in its closed neighborhood. In order to minimize the risk of intercepting the information, we are interested in selecting a secluded path, i.e. a path with a small exposure. Similarly, the Secluded Steiner Tree problem is to find a tree in a graph connecting a given set of terminals such that the exposure of the tree is minimized. In this work, we obtain the following results about parameterized complexity of secluded connectivity problems. We start from an observation that being parameterized by the size of the exposure, the problem is fixed-parameter tractable (FPT). More precisely, we give an algorithm deciding if a graph G with a given cost function w:V(G)->N contains a secluded path of exposure at most k with the cost at most C in time O(3^{k/3}(n+m) log W), where W is the maximum value of w on an input graph G. Similarly, Secluded Steiner Tree is solvable in time O(2^{k}k^2 (n+m) log W). The main result of this paper is about "above guarantee" parameterizations for secluded problems. We show that Secluded Steiner Tree is FPT being parameterized by r+p, where p is the number of the terminals, l the size of an optimum Steiner tree, and r=k-l. We complement this result by showing that the problem is co-W[1]-hard when parameterized by r only. We also investigate Secluded Steiner Tree from kernelization perspective and provide several lower and upper bounds when parameters are the treewidth, the size of a vertex cover, maximum vertex degree and the solution size. Finally, we refine the algorithmic result of Chechik et al. by improving the exponential dependence from the treewidth of the input graph.
Cite as

Fedor V. Fomin, Petr A. Golovach, Nikolay Karpov, and Alexander S. Kulikov. Parameterized Complexity of Secluded Connectivity Problems. In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. 408-419, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{fomin_et_al:LIPIcs.FSTTCS.2015.408,
  author =	{Fomin, Fedor V. and Golovach, Petr A. and Karpov, Nikolay and Kulikov, Alexander S.},
  title =	{{Parameterized Complexity of Secluded Connectivity Problems}},
  booktitle =	{35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)},
  pages =	{408--419},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-97-2},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{45},
  editor =	{Harsha, Prahladh and Ramalingam, G.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.408},
  URN =		{urn:nbn:de:0030-drops-56318},
  doi =		{10.4230/LIPIcs.FSTTCS.2015.408},
  annote =	{Keywords: Secluded path, Secluded Steiner tree, parameterized complexity}
}
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