Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Morawietz, Nils; Grüttemeier, Niels; Komusiewicz, Christian; Sommer, Frank https://www.dagstuhl.de/lipics License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
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URN: urn:nbn:de:0030-drops-132719
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Colored Cut Games

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Abstract

In a graph G = (V,E) with an edge coloring 𝓁:E → C and two distinguished vertices s and t, a colored (s,t)-cut is a set C̃ ⊆ C such that deleting all edges with some color c ∈ C̃ from G disconnects s and t. Motivated by applications in the design of robust networks, we introduce a family of problems called colored cut games. In these games, an attacker and a defender choose colors to delete and to protect, respectively, in an alternating fashion. It is the goal of the attacker to achieve a colored (s,t)-cut and the goal of the defender to prevent this. First, we show that for an unbounded number of alternations, colored cut games are PSPACE-complete. We then show that, even on subcubic graphs, colored cut games with a constant number i of alternations are complete for classes in the polynomial hierarchy whose level depends on i. To complete the dichotomy, we show that all colored cut games are polynomial-time solvable on graphs with degree at most two. Finally, we show that all colored cut games admit a polynomial kernel for the parameter k+κ_r where k denotes the total attacker budget and, for any constant r, κ_r is the number of vertex deletions that are necessary to transform G into a graph where the longest path has length at most r. In the case of r = 1, κ₁ is the vertex cover number vc of the input graph and we obtain a kernel with 𝒪(vc²k²) edges. Moreover, we introduce an algorithm solving the most basic colored cut game, Colored (s,t)-Cut, in 2^{vc + k}n^{𝒪(1)} time.

BibTeX - Entry

@InProceedings{morawietz_et_al:LIPIcs:2020:13271,
  author =	{Nils Morawietz and Niels Gr{\"u}ttemeier and Christian Komusiewicz and Frank Sommer},
  title =	{{Colored Cut Games}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{30:1--30:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Nitin Saxena and Sunil Simon},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/13271},
  URN =		{urn:nbn:de:0030-drops-132719},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.30},
  annote =	{Keywords: Labeled Cut, Labeled Path, Network Robustness, Kernelization, PSPACE, Polynomial Hierarchy}
}

Keywords: Labeled Cut, Labeled Path, Network Robustness, Kernelization, PSPACE, Polynomial Hierarchy
Seminar: 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)
Issue date: 2020
Date of publication: 04.12.2020


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