A Polynomial Time Algorithm for Spatio-Temporal Security Games

TL;DR

This paper presents a polynomial-time algorithm for computing Nash equilibria in a class of spatio-temporal security games, significantly improving computational efficiency over prior methods.

Contribution

It introduces the first polynomial-time algorithm for these security games that is efficient even with a large number of patrol locations and extends to continuous patrol positioning.

Findings

Algorithm runs in polynomial time in input size

Efficient even with large number of patrol locations

Supports continuous patrol location values

Abstract

An ever-important issue is protecting infrastructure and other valuable targets from a range of threats from vandalism to theft to piracy to terrorism. The "defender" can rarely afford the needed resources for a 100% protection. Thus, the key question is, how to provide the best protection using the limited available resources. We study a practically important class of security games that is played out in space and time, with targets and "patrols" moving on a real line. A central open question here is whether the Nash equilibrium (i.e., the minimax strategy of the defender) can be computed in polynomial time. We resolve this question in the affirmative. Our algorithm runs in time polynomial in the input size, and only polylogarithmic in the number of possible patrol locations (M). Further, we provide a continuous extension in which patrol locations can take arbitrary real values. Prior work obtained polynomial-time algorithms only under a substantial assumption, e.g., a constant number of rounds. Further, all these algorithms have running times polynomial in M, which can be very large.