infrastructure protection planners

Abstract: In a repeated Stackelberg security game, a defender commits in every round to a randomized allocation of its resources, and an attacker, drawn from a set of types, best-responds. Existing no-regret guarantees assume that the defender knows this set of types in advance. We drop this assumption: the set of types is unknown, and its size is at most $K_{\max}$, an upper bound that the defender is not told. We show that what can be learned then depends on what the defender observes. If it observes the type of each attacker, an algorithm that adds types to its list as they appear achieves regret $\tilde O(nK_{\max}\sqrt{T})$ over $T$ rounds with $n$ targets, the same $\sqrt T$ rate as with known types, at an extra factor of only $\sqrt{K_{\max}}$ and without ever using $K_{\max}$. If it observes only the attacked target and types come from an infinite universe, no algorithm has sublinear regret: in a fixed three-target game every algorithm suffers regret at least $T-2$, even against a single attacker type, because it cannot tell which type it faces. If, in addition, it is told every $B$ rounds which types have appeared so far, sublinear regret returns, at $\tilde O(nK_{\max}^{4/3}T^{2/3}+K_{\max} B)$. The first term is the rate of the partial-information learner we use, and we show that the additive cost of the delay cannot be avoided.

Abstract: Eternal Vertex Cover (EVC) is a turn-based attacker-defender game on an undirected graph $G$. To begin with, the defender places $k$ guards on vertices of $G$. The attacker, on their turn, can choose an edge $e$ not already occupied at both endpoints to "attack". The edge $e$ is defended if a guard moves along the edge $e$. The defender, on their turn, can move any subset of guards. A guard can only move to a neighboring vertex. The minimum number of guards needed to indefinitely defend against any sequence of attacks is called the eternal vertex cover number, generalizing the classic vertex cover number. Determining this number is NP-hard in general, motivating the study of parameterized and approximation algorithms. The problem is known to be FPT when parameterized by the cover number, but structural parameters remain relatively unexplored in the literature. In this work, we explore structural parameterizations for EVC. We show that EVC is FPT parameterized by the cluster vertex deletion number, which generalizes the previously studied parameterization by vertex cover number. We next study the problem parameterized by vertex integrity, which is the smallest number of vertices we need to delete from $G$ so that the resulting graph is a disjoint union of constant-sized components. We first show that Eternal Vertex Cover is XP parameterized by vertex integrity. Then, we develop a polynomial-time approximation algorithm, which computes an additive $6k+1$ ($g(k)$) approximation, where $k$ is equal to the cluster vertex deletion number (vertex integrity). Finally, we show a FPT algorithm for when the deletion set produces "nice" connected components, which are components that are bounded in size and satisfy a technical condition.