A Little Aggression Goes a Long Way
Jyothi Krishnan, Neeldhara Misra, Saraswati Girish Nanoti
TL;DR
The paper studies Aggression, a two-phase graph game with a placement phase followed by an attack phase, and proves that computing an optimal first-player response is NP-complete even under a fixed second-player plan. It provides a detailed reduction from Multi-Colored Clique to Optimal Response and shows the construction can be made bipartite and connected, establishing hardness. It also characterizes outcomes on restricted graphs, proving draw strategies on matchings of small size and certain edge/troop relationships, and showing draw results on cycles of length at most five. These results illuminate the computational complexity of strategic planning in graph-based pursuit-and-attack games and suggest directions for future work on planar/acyclic cases and Micro Aggression variants.
Abstract
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.