eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-12-04
30:1
30:17
10.4230/LIPIcs.FSTTCS.2020.30
article
Colored Cut Games
Morawietz, Nils
1
Grüttemeier, Niels
1
https://orcid.org/0000-0002-6789-2918
Komusiewicz, Christian
1
https://orcid.org/0000-0003-0829-7032
Sommer, Frank
1
https://orcid.org/0000-0003-4034-525X
Philipps-Universität Marburg, Fachbereich Mathematik und Informatik, Germany
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol182-fsttcs2020/LIPIcs.FSTTCS.2020.30/LIPIcs.FSTTCS.2020.30.pdf
Labeled Cut
Labeled Path
Network Robustness
Kernelization
PSPACE
Polynomial Hierarchy