Colored cut games

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 C˜⊆C such that deleting all edges with some color c∈C˜ from G disconnects s and t. Motivated by applications in the design of robust networks, we introduce colored cut games. In these games, an attacker and a defender choose colors to delete and to protect, respectively, in an alternating fashion. The attacker wants to achieve a colored (s,t)-cut and the defender wants to prevent this. First, we show that for an unbounded number of alternations, colored cut games are PSPACE-complete even on subcubic graphs. We then show that, even on subcubic graphs, colored cut games with i 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 maximum degree at most 2.Next, 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. For κ1, which is the vertex cover number vc of the input graph, the kernel has size O(vc2k2). Moreover, we introduce an algorithm solving the most basic colored cut game, Colored(s,t)-Cut, in 2vc+knO(1) time.