The Minimum Backlog Problem

We study the minimum backlog problem (MBP). This online problem arises, e.g., in the context of sensor networks. We focus on two main variants of MBP. The discrete MBP is a 2-person game played on a graph $G=(V,E)$. The player is initially located at a vertex of the graph. In each time step, the adversary pours a total of one unit of water into cups that are located on the vertices of the graph, arbitrarily distributing the water among the cups. The player then moves from her current vertex to an adjacent vertex and empties the cup at that vertex. The player's objective is to minimize the backlog, i.e., the maximum amount of water in any cup at any time. The geometric MBP is a continuous-time version of the MBP: the cups are points in the two-dimensional plane, the adversary pours water continuously at a constant rate, and the player moves in the plane with unit speed. Again, the player's objective is to minimize the backlog. We show that the competitive ratio of any algorithm for the MBP has a lower bound of $\Omega(D)$, where $D$ is the diameter of the graph (for the discrete MBP) or the diameter of the point set (for the geometric MBP). Therefore we focus on determining a strategy for the player that guarantees a uniform upper bound on the absolute value of the backlog. For the absolute value of the backlog there is a trivial lower bound of $\Omega(D)$, and the deamortization analysis of Dietz and Sleator gives an upper bound of $O(D\log N)$ for $N$ cups. Our main result is a tight upper bound for the geometric MBP: we show that there is a strategy for the player that guarantees a backlog of $O(D)$, independently of the number of cups.Comment: 1+16 pages, 3 figure

Robot Planning in Adversarial Environments Using Tree Search Techniques

One of the main advantages of robots is that they can be used in environments that are dangerous for humans. Robots can not only be used for tasks in known and safe areas but also in environments that may have adversaries. When planning the robot's actions in such scenarios, we have to consider the outcomes of a robot's actions based on the actions taken by the adversary, as well as the information available to the robot and the adversary. The goal of this dissertation is to design planning strategies that improve the robot's performance in adversarial environments. Specifically, we study how the availability of information affects the planning process and the outcome. We also study how to improve the computational efficiency by exploiting the structural properties of the underlying setting. We adopt a game-theoretic formulation and study two scenarios: adversarial active target tracking and reconnaissance in environments with adversaries. A conservative approach is to plan the robot's action assuming a worst-case adversary with complete knowledge of the robot's state and objective. We start with such a "symmetric" information game for the adversarial target tracking scenario with noisy sensing. By using the properties of the Kalman filter, we design a pruning strategy to improve the efficiency of a tree search algorithm. We investigate the performance limits of the asymmetric version where the adversary can inject false sensing data. We then study a reconnaissance scenario where the robot and the adversary have symmetric information. We design an algorithm that allows a robot to scan more area while avoiding being detected by the adversary. The symmetric adversarial model may yield too conservative plans when the adversary may not have the same information as the robot. Furthermore, the information available to the adversary may change during execution. We then investigate the dynamic version of this asymmetric information game and show how much the robot can exploit the asymmetry in information using tree search techniques. Specifically, we study scenarios where the information available to the adversary changes during execution. We devise a new algorithm for this asymmetric information game with theoretical performance guarantees and evaluate those approaches through experiments. We use qualitative examples to show how the new algorithm can outperform symmetric minimax and use quantitative experiments to show how much the improvement is

Geometric Pursuit Evasion

In this dissertation we investigate pursuit evasion problems set in geometric environments. These games model a variety of adversarial situations in which a team of agents, called pursuers, attempts to catch a rogue agent, called the evader. In particular, we consider the following problem: how many pursuers, each with the same maximum speed as the evader, are needed to guarantee a successful capture? Our primary focus is to provide combinatorial bounds on the number of pursuers that are necessary and sufficient to guarantee capture. The first problem we consider consists of an unpredictable evader that is free to move around a polygonal environment of arbitrary complexity. We assume that the pursuers have complete knowledge of the evader's location at all times, possibly obtained through a network of cameras placed in the environment. We show that regardless of the number of vertices and obstacles in the polygonal environment, three pursuers are always sufficient and sometimes necessary to capture the evader. We then consider several extensions of this problem to more complex environments. In particular, suppose the players move on the surface of a 3-dimensional polyhedral body; how many pursuers are required to capture the evader? We show that 4 pursuers always suffice (upper bound), and that 3 are sometimes necessary (lower bound), for any polyhedral surface with genus zero. Generalizing this bound to surfaces of genus g, we prove the sufficiency of (4g + 4) pursuers. Finally, we show that 4 pursuers also suffice under the "weighted region" constraints, where the movement costs through different regions of the (genus zero) surface have (different) multiplicative weights. Next we consider a more general problem with a less restrictive sensing model. The pursuers' sensors are visibility based, only providing the location of the evader if it is in direct line of sight. We begin my making only the minimalist assumption that pursuers and the evader have the same maximum speed. When the environment is a simply-connected (hole-free) polygon of n vertices, we show that Θ(n^1/2 ) pursuers are both necessary and sufficient in the worst-case. When the environment is a polygon with holes, we prove a lower bound of Ω(n^2/3 ) and an upper bound of O(n^5/6 ) pursuers, where n includes the vertices of the hole boundaries. However, we show that with realistic constraints on the polygonal environment these bounds can be drastically improved. Namely, if the players' movement speed is small compared to the features of the environment, we give an algorithm with a worst case upper bound of O(log n) pursuers for simply-connected n-gons and O(√h + log n) for polygons with h holes. The final problem we consider takes a small step toward addressing the fact that location sensing is noisy and imprecise in practice. Suppose a tracking agent wants to follow a moving target in the two-dimensional plane. We investigate what is the tracker's best strategy to follow the target and at what rate does the distance between the tracker and target grow under worst-case localization noise. We adopt a simple but realistic model of relative error in sensing noise: the localization error is proportional to the true distance between the tracker and the target. Under this model we are able to give tight upper and lower bounds for the worst-case tracking performance, both with or without obstacles in the Euclidean plane