A Little Aggression Goes a Long Way

Aggression is a two-player game of troop placement and attack played on a map (modeled as a graph). Players take turns deploying troops on a territory (a vertex on the graph) until they run out. Once all troops are placed, players take turns attacking enemy territories. A territory can be attacked if it has $k$ troops and there are more than $k$ enemy troops on adjacent territories. At the end of the game, the player who controls the most territories wins. In the case of a tie, the player with more surviving troops wins. The first player to exhaust their troops in the placement phase leads the attack phase. We study the complexity of the game when the graph along with an assignment of troops and the sequence of attacks planned by the second player. Even in this restrained setting, we show that the problem of determining an optimal sequence of first player moves is NP-complete. We then analyze the game for when the input graph is a matching or a cycle.