Colored Cut Games

In a graph G = (V,E) with an edge coloring ` : 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 + κr where k denotes the total attacker budget and, for any constant r, κr 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 2vc+knO(1) time. 2012 ACM Subject Classification Theory of computation → Parameterized complexity and exact algorithms; Theory of computation → Graph algorithms analysis; Theory of computation → Problems, reductions and completeness