Branch-and-price with novel cuts, and a new Stackelberg Security Game
Pamela Bustamante-Faúndez, Víctor Bucarey L., Martine Labbé, Vladimir Marianov, Fernando Ordóñez
TL;DR
This work tackles defending critical infrastructure by modeling defender–attacker interactions as Stackelberg security games (SSG) and Bayesian Stackelberg games, with a focus on Strong Stackelberg Equilibrium (SSE). It introduces two strengthened formulations, $(\text{D2}_+)$ and $(\text{D2}_+^{\prime})$, equipped with new valid inequalities that tighten the MILP relaxations for Stackelberg games and Stackelberg security games. A noncompact, budget-aware problem is solved via a branch-and-price approach, using a knapsack-like column-generation master problem and a specialized pricing problem; Farkas pricing is incorporated to handle infeasibility, and heuristic pricers (GRASP, Ratio Greedy) are employed to accelerate convergence. Computational experiments show large speedups and better LP gaps for the budgeted SSG, with one-column-per-iteration pricing and combined defender/attacker inequalities yielding the best performance, outperforming the standard $(\text{D2})$ formulation and a Benders approach. The results demonstrate the practical potential of branch-and-price with new cuts for large-scale, multi-attacker security problems and budget-constrained defense planning.
Abstract
Anticipating the strategies of potential attackers is crucial for protecting critical infrastructure. We can represent the challenge of the defenders of such infrastructure as a Stackelberg security game. The defender must decide how to allocate limited resources to protect specific targets, aiming to maximize their expected utility (such as minimizing the extent of damage) and considering that attackers will respond in a way that is most advantageous to them. We present novel valid inequalities to find a Strong Stackelberg Equilibrium in both Stackelberg games and Stackelberg security games. We also consider a Stackelberg security game that aims to protect targets with a defined budget. We use branch-and-price in this game to show that our approach outperforms the standard formulation in the literature, and we conduct an extensive computational study to analyze the impact of various branch-and-price parameters on the performance of our method in different game settings.