Faster Algorithms for Security Games on Matroids

Abstract

Given a matroid M defined by an independence oracle on a ground set E, the Matroid Base Game is played by two players: the defender chooses a basis B and (simultaneously) the attacker chooses an element $e \in E$. The attacker incurs a cost c(e) for choosing an element e, and if $e \in B$ then there is a probability p(e) that the attacker will detect the defender. The defender has to find a bases-selection strategy that minimizes the average probability of being detected. The attacker has to find a probabilistic selection strategy that maximizes the average detection probability minus its average cost. An algorithm to compute Nash-equilibrium mixed strategies was given in Szeszlér (Math Program 161:347–364, 2016). Its time complexity is $O(|E|^{10} IO)$, where IO is the time that it takes one call to the independence oracle. Here we present an algorithm that requires $O(|E|^6 IO)$ time. For graphic matroids, i.e., when the defender chooses a spanning tree in a graph $G=(V,E)$, and the attacker chooses an edge, we give an algorithm that takes $O(|V|^4 |E|^{1/2})$ time. This type of game is extended to common bases of two matroids. For this case we give a strongly polynomial algorithm, settling a question that was left open in Szeszlér (2016). We also treat the case when the defender chooses a rooted arborescence in a directed graph $D=(V,\mathscr {A})$, and the attacker chooses an arc, we use this structure to give an algorithm that requires $O(|V||\mathscr {A}|^3 \log (|V|^2/|\mathscr {A}|) \log |\mathscr {A}|)$ time.