Simplifying Urban Network Security Games with Cut-Based Graph Contraction

The scalability of the algorithm for solving urban network security games, which is an important challenge concerning security game problems, was improved. State-of-the-art solvers have been scaled up to handle real-world networks with tens of thousands of edges; however, it can take days or more when the inputs are varied. Since they do not essentially overcome exponential growth of the strategy space with increasing graph size, an approach, which can be combined with previous ones, is proposed. In particular, a practical approach of simplifying the graphs so that they can be handled within a realistic time is devised and tested. The key idea behind this approach is to restrict the defender's pure strategies to potential ones before calculating an equilibrium solution. The restriction can be tightened for faster computation and loosened for better solution. The following three techniques for computing an optimal solution to the restricted game are proposed and evaluated: (i) contraction of the network based on the restriction, (ii) compact formulation of the optimization problem using weighted edges in place of multiple edges, and (iii) efficient solution using a mixed-integer quadratic programming oracle. They can naturally cope with an extension of the game to one taking width of the roads into account. Furthermore, a heuristic algorithm of finding effective restriction of the game is also proposed.