Spatio-Temporal Games Beyond One Dimension

Protecting valuable \em targets from an adversary is an ever-important international concern with far-reaching applications in wildlife protection, border protection, counter-terrorism, protection of ships from piracy, etc. As a successful recent approach, \em security games cast these issues as two-player games between a \em defender and an \em attacker. The defender decides on how to allocate the available \em resources to protect targets against the attacker who strives to inflict damage on them. The main question of interest here is equilibrium computation. Our focus in this paper is on \em spatio-temporal security games. However, inspired by the paper of Xu [EC'16], we start with a general model of security games and show that any approximation (of any factor) for the defender's best response (DBR) problem leads to an approximation of the same factor for the actual game. In most applications of security games, the targets are mobile. This leads to a well-studied class of succinct games, namely \em spatio-temporal security games, that is played in space and time. In such games, the defender has to specify a time-dependent patrolling strategy over a spatial domain to protect a set of moving targets. We give a generalized model of prior spatio-temporal security games that is played on a base graph G . That is, the patrols can be placed on the vertices of G and move along its edges over time. This unifies and generalizes prior spatio-temporal models that only consider specific spatial domains such as lines or grids. Graphs can further model many other domains of practical interest such as roads, internal maps of buildings, etc. Finding an optimal defender strategy becomes NP-hard on general graphs. To overcome this, we give an LP relaxation of the DBR problem and devise a rounding technique to obtain an almost optimal integral solution. More precisely, we show that one can achieve a $(1-ε)$-approximation in polynomial time if we allow the defender to use $łceil łn(1/ε)\rceil$ times more patrols. We later show that this result is in some sense the best possible polynomial time algorithm (unless P=NP). Furthermore, we show that by using a novel \em dependent rounding technique, the same LP relaxation gives an optimal solution for specific domains of interest, such as one-dimensional spaces. This result simplifies and improves upon the prior algorithm of Behnezhad et al. ~[EC'17] on several aspects and can be generalized to other graphs of interest such as cycles. Lastly, we note that most prior algorithms for security games assume that the attacker attacks only once and become intractable for a super-constant number of attacks. Our algorithms are fully polynomial in the input size and work for any given number of attacks.