Colored Cut Games
In a graph G = (V,E) with an edge coloring đ:E â C and two distinguished vertices s and t, a colored (s,t)-cut is a set CÌ â C such that deleting all edges with some color c â CÌ 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+Îș_r where k denotes the total attacker budget and, for any constant r, Îș_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, Îșâ is the vertex cover number vc of the input graph and we obtain a kernel with đȘ(vcÂČkÂČ) edges. Moreover, we introduce an algorithm solving the most basic colored cut game, Colored (s,t)-Cut, in 2^{vc + k}n^{đȘ(1)} time.
Labeled Cut
Labeled Path
Network Robustness
Kernelization
PSPACE
Polynomial Hierarchy
Theory of computation~Parameterized complexity and exact algorithms
Theory of computation~Graph algorithms analysis
Theory of computation~Problems, reductions and completeness
30:1-30:17
Regular Paper
Some of the results of this work are also contained in the first authorâs Master thesis [Nils Morawietz, 2019].
Nils
Morawietz
Nils Morawietz
Philipps-UniversitĂ€t Marburg, Fachbereich Mathematik und Informatik, Germany
{Partially supported by the Deutsche Forschungsgemeinschaft (DFG), project OPERAH, KO 3669/5-1.}
Niels
GrĂŒttemeier
Niels GrĂŒttemeier
Philipps-UniversitĂ€t Marburg, Fachbereich Mathematik und Informatik, Germany
https://orcid.org/0000-0002-6789-2918
Christian
Komusiewicz
Christian Komusiewicz
Philipps-UniversitĂ€t Marburg, Fachbereich Mathematik und Informatik, Germany
https://orcid.org/0000-0003-0829-7032
Frank
Sommer
Frank Sommer
Philipps-UniversitĂ€t Marburg, Fachbereich Mathematik und Informatik, Germany
https://orcid.org/0000-0003-4034-525X
Supported by the Deutsche Forschungsgemeinschaft (DFG), project MAGZ, KO 3669/4-1.
10.4230/LIPIcs.FSTTCS.2020.30
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Nils Morawietz, Niels GrĂŒttemeier, Christian Komusiewicz, and Frank Sommer
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