Introduction

In this chapter, we consider a more complicated scheduling problem than Chapter 15, where the resource needed for the processing of packets, energy, itself arrives over time, and the algorithm has only causal knowledge about it. This paradigm is relevant for communication systems powered by renewable energy sources, where the amount of energy arriving at each time slot is unpredictable. This scenario also models scheduling problems on factory floors where the availability time of, say, raw materials or multiple machines needed to complete a complicated job is uncertain and is revealed causally to the algorithm.

Conventionally, in online scheduling, unlimited energy is available, and the objective is to minimize a combination of energy used and relevant performance metrics, e.g., makespan, completion time, flow time, and the only uncertainty is about the packet or job arrival times and their sizes. For the online scheduling problem considered in the chapter, both the amount of energy and its arrival slots in the future are unknown to the algorithm and are possibly controlled by the adversary. With energy arriving over time and in arbitrary amounts, the generic scheduling problem is to minimize any performance metric, subject to the energy neutrality constraint, i.e., the amount of energy used by any time is at most the amount of energy that has arrived so far.

With arbitrary energy arrivals, we consider a canonical problem of transmitting a single packet (with a fixed number of bits) to minimize its completion time. Without loss of generality, we assume that the total amount of energy that arrives over time is sufficient to transmit the packet completely by the optimal offline algorithm OPT. The challenge is to propose an algorithm that can compete with OPT. We show that a Lazy algorithm has a competitive ratio of 2, which is also the best possible.

The considered problem in this chapter is fairly versatile. For example, it can model a scheduling paradigm where there are multiple servers but their availability is unknown, and the amount of work done in each slot is a concave function of the number of servers used in that slot. Then a scheduling problem (with any usual performance metric such as flow time, makespan, completion time) emerges where the decision is: how many servers to use among the available ones at each time.

Chapter 3 introduces the concept of the rule base along with the material from Chapter 2 to construct different fuzzy inference systems such as the Mamdani fuzzy inference system or the Sugeno fuzzy inference system. The Takagi-Sugeno fuzzy inference system is used to design fuzzy logic controllers and Lyapunov theory is utilized to investigate the closed-loop system stability of such controllers. Concepts such as local sector nonlinearity, globally asymptotical stability using state-space models are introduced and discussed to fashion controllers for nonlinear systems. Throughout the chapter, Matlab’s FIS editor is used to design fuzzy inference systems and corresponding controllers.

Starting with crisp set theory, fuzzy sets and concepts of fuzzy logic are introduced in Chapter 2. Some of the key operators are discussed and utilized in a number of examples. Membership functions, membership operators, their programming in Matlab, as well as logic operators using membership functions are explained. Along with conditional statements such as fuzzy rules and linguistic variables concepts such as antecedents, consequences and inference are discussed and shown how to implement this type of reasoning in Matlab.

Chapter 1 provides for an introduction to the key concepts of the book, including supervised and unsupervised learning, reinforcement learning, and controls. The objective is to provide an overview of the many methods and algorithms and how they are relate to each other as well as to controls applications.

Based on Chapter 6, in this chapter we expand the discussion of neural networks to include networks that have more than one hidden layer. Common structures such as the convolutional neural network (CNN) or the Long Short-Term Memory network (LSTM) are explained and used along with Matlab’s Deep Network Designer App as well as Matlab script to implement and train such networks. Issues such as the vanishing or exploding gradient, normalization, and training strategies are discussed. Concepts that address overfitting and the vanishing or exploding gradient are introduced, including dropout and regularization. Transfer learning is discussed and showcased using Matlab’s DND App.

Introduction

In this chapter, we visit a classical combinatorial problem, the travelling salesman problem (TSP). In the offline case, TSP is formulated over an undirected graph, where each edge has a weight, and the objective is to minimize the total edge weight of the tour that starts and ends at the same vertex and visits each vertex of the graph at least once. TSP in the offline case is a very rich problem and has been an object of intense study. In the online setting, TSP can be posed in multiple ways, and we study two of the most prominent versions in this chapter.

The first version we consider involves sites to be visited that belong to a metric space. Consider a walker that can walk at most unit speed. Starting at a fixed site, sites (locations) to be visited in the future belonging to a metric space are revealed sequentially, while the walker is travelling. The goal of the walker is to visit all sites and return to the starting site in the minimum time possible while ensuring that a site is visited only after it has been revealed. This version captures some of the online counterparts of the usual offline TSP applications. For this version, we first show that the competitive ratio of any online algorithm is at least 2, and then present a simple algorithm that achieves the lower bound.

The second version we consider is an exploration problem over an unknown graph. Assume that a walker is at a particular (starting) vertex of an unknown undirected edge weighted graph G. The walker's objective is to visit all the vertices of G and return to the starting vertex over the shortest path. The online restriction is that at any time, only the neighbours of all the visited vertices so far and the associated edge weights are revealed. Thus, each time a walker reaches a new vertex, it has to decide which vertex to visit next, given the partial graph information. This unknown graph exploration version is seemingly more difficult than the site exploration version, and we present the best-known online algorithm with the competitive ratio of at most 16.

Chapter 8 explores the application of international law and norms in cyberspace. It examines law that governs use of force in international politics, the types of weapons that can be developed and used in armed conflict, how combatants engage in conflict on the battlefield, and when individuals can be held criminally accountable for violating these rules. Some governments classify a cyber operation as equivalent to a use of force or form of warfare only if it produces physical destruction or death, while others declare a cyber operation to be an attack when the targeted system loses its ability to function. Finally, the chapter considers how to develop and encourage the adoption of cyberspace norms for governmental behavior and set expectations for states to regulate illicit cyber activity within their borders.

Introduction

In this chapter, we discuss the online version of one of the most versatile combinatorial problems, called the knapsack. In the classical offline version of the knapsack problem, we are provided with a set of items I and a knapsack of size or capacity C. Each item i * I has value v(i) and size w(i), and the problem is to select a subset of I that maximizes the sum of the value of the selected items, subject to the sum of the sizes of the selected items being less than the capacity C of the knapsack.

Because of the two unrelated attributes for each item, value and size, the knapsack problem is sufficient to model various real-world problems where the objective is to maximize a utility function subject to an independent capacity constraint. Important examples of the knapsack problem are scheduling with resource capacity constraints, budgeted auctions, combinatorial resource allocation, etc.

In the online version, the knapsack capacity constraint is available ahead of time, but each item is presented sequentially when its value and size are revealed. An item on its arrival has to be permanently accepted or rejected, irrevocably. It is worth mentioning that the secretary problem considered in Chapter 7 is a special case of the online knapsack problem, where the size of each item is 1 and the knapsack capacity is also 1. Thus, unfortunately, the result that no algorithm is competitive for the secretary problem under the adversarial input carries over for the knapsack problem. Therefore, in this chapter, we primarily consider the secretarial input and present an online algorithm whose competitive ratio is a constant.

We show that a randomized algorithm based on the sample and price philosophy is 1/10ecompetitive in expectation for the knapsack problem with the secretarial input model. We also consider the worst-case input, though with resource augmentation, where an online algorithm is allowed more capacity than the optimal offline algorithm, and an online algorithm is also allowed to reject previously accepted items.

Chapter 7 focuses on the impact of international cooperation in cyberspace. By design, the Internet is global, and engineering ignores sovereignty concerns such as citizenship, borders, and domestic law. However, running against domestic regulation and the consequent possible internet fragmentation are several international efforts to cooperate in cyberspace through various stakeholder and multilateral models that include the United Nations, European Union, and the North Atlantic Treaty Organization (NATO). The latter, composed of 30 countries in North America and Europe, issued a communique in 2021 pushing back against authoritarian uses of the Internet calling for a free, open, and peaceful cyberspace. In 2022, 60 countries, including many NATO members, adopted ”A Declaration for the Future of the Internet.” Applying human rights in cyberspace was among the principles adopted. To explore these issues, the chapter reviews forms of internet governance enabling technical, legal, and policy cooperation across boundaries.