This book starts by providing the relevant background on computability theory, which is the setting in which Complexity theoretic questions are being studied. Most importantly, this preliminary chapter (i.e., Chapter 1) provides a treatment of central notions, such as search and decision problems, algorithms that solve such problems, and their complexity. Special attention is given to the notion of a universal algorithm.

The main part of this book (i.e., Chapters 2–5) focuses on the P-vs-NP Question and on the theory of NP-completeness. Additional topics covered in this part include the general notion of an efficient reduction (with a special emphasis on reductions of search problems to corresponding decision problems), the existence of problems in NP that are neither NP-complete nor in P, the class coNP, optimal search algorithms, and promise problems. A brief overview of this main part follows.

The P-vs-NP Question. Loosely speaking, the P-vs-NP Question refers to search problems for which the correctness of solutions can be efficiently checked (i.e., there is an efficient algorithm that given a solution to a given instance determines whether or not the solution is correct). Such search problems correspond to the class NP, and the P-vs-NP Question corresponds to whether or not all these search problems can be solved efficiently (i.e., is there an efficient algorithm that given an instance finds a correct solution). Thus, the P-vs-NP Question can be phrased as asking whether finding solutions is harder than checking the correctness of solutions.

The following brief overview is intended to give a flavor of the questions addressed by Complexity Theory. It includes a brief review of the contents of the current book, as well as a brief overview of several more advanced topics. The latter overview is quite vague, and is merely meant as a teaser toward further study (cf., e.g., [13]).

Absolute Goals and Relative Results

Complexity Theory is concerned with the study of the intrinsic complexity of computational tasks. Its “final” goals include the determination of the complexity of any well-defined task. Additional goals include obtaining an understanding of the relations between various computational phenomena (e.g., relating one fact regarding Computational Complexity to another). Indeed, we may say that the former type of goals is concerned with absolute answers regarding specific computational phenomena, whereas the latter type is concerned with questions regarding the relation between computational phenomena.

Interestingly, so far Complexity Theory has been more successful in coping with goals of the latter (“relative”) type. In fact, the failure to resolve questions of the “absolute” type led to the flourishing of methods for coping with questions of the “relative” type. Musing for a moment, let us say that, in general, the difficulty of obtaining absolute answers may naturally lead to a search for conditional answers, which may in turn reveal interesting relations between phenomena. Furthermore, the lack of absolute understanding of individual phenomena seems to facilitate the development of methods for relating different phenomena. Anyhow, this is what happened in Complexity Theory.

In this chapter we discuss three relatively advanced topics. The first topic, which was alluded to in previous chapters, is the notion of promise problems (Section 5.1). Next, we present an optimal algorithm for solving (“candid”) NP-search problems (Section 5.2). Finally, in Section 5.3, we briefly discuss the class (denoted coNP) of sets that are complements of sets in NP.

Teaching Notes

Typically, the foregoing topics are not mentioned in a basic course on complexity. Still, we believe that these topics deserve at least a mention in such a course. This holds especially with respect to the notion of promise problems. Furthermore, depending on time constraints, we recommend presenting all three topics in class (at least at an overview level).

We comment that the notion of promise problems was originally introduced in the context of decision problems, and is typically used only in that context. However, given the importance that we attach to an explicit study of search problems, we extend the formulation of promise problems to search problems as well. In that context, it is also natural to introduce the notion of a “candid search problem” (see Definition 5.2).

Promise Problems

Promise problems are natural generalizations of search and decision problems. These generalizations are obtained by explicitly considering a set of legitimate instances (rather than considering any string as a legitimate instance). As noted previously, this generalization provides a more adequate formulation of natural computational problems (and, indeed, this formulation is used in all informal discussions).

Although we view specific (natural) computational problems as secondary to (natural) complexity classes, we do use the former for clarification and illustration of the latter. This appendix provides definitions of such computational problems, grouped according to the type of objects to which they refer (i.e., graphs and Boolean formula).

We start by addressing the central issue of the representation of the various objects that are referred to in the aforementioned computational problems. The general principle is that elements of all sets are “compactly” represented as binary strings (without much redundancy). For example, the elements of a finite set S (e.g., the set of vertices in a graph or the set of variables appearing in a Boolean formula) will be represented as binary strings of length log2 |S|.

Graphs

A simple graph G=(V,E) consists of a finite set of vertices V and a finite set of edges E, where each edge is an unordered pair of vertices; that is, E ⊆ {{u, v} : u, v∈V ∧ u≠v}.

According to a common opinion, the most important aspect of a scientific work is the technical result that it achieves, whereas explanations and motivations are merely redundancy introduced for the sake of “error correction” and/or comfort. It is further believed that, as with a work of art, the interpretation of the work should be left to the reader.

The author strongly disagrees with the aforementioned opinions, and argues that there is a fundamental difference between art and science, and that this difference refers exactly to the meaning of a piece of work. Science is concerned with meaning (and not with form), and in its quest for truth and/or understanding, science follows philosophy (and not art). The author holds the opinion that the most important aspects of a scientific work are the intuitive question that it addresses, the reason that it addresses this question, the way it phrases the question, the approach that underlies its answer, and the ideas that are embedded in the answer. Following this view, it is important to communicate these aspects of the work.

The foregoing issues are even more acute when it comes to Complexity Theory, firstly because conceptual considerations seem to play an even more central role in Complexity Theory than in other scientific fields. Secondly (and even more importantly), Complexity Theory is extremely rich in conceptual content. Thus, communicating this content is of primary importance, and failing to do so misses the most important aspects of Complexity Theory.

The quest for efficiency is ancient and universal, as time and other resources are always in shortage. Thus, the question of which tasks can be performed efficiently is central to the human experience.

A key step toward the systematic study of the aforementioned question is a rigorous definition of the notion of a task and of procedures for solving tasks. These definitions were provided by computability theory, which emerged in the 1930s. This theory focuses on computational tasks, considers automated procedures (i.e., computing devices and algorithms) that may solve such tasks, and studies the class of solvable tasks.

In focusing attention on computational tasks and algorithms, computability theory has set the stage for the study of the computational resources (like time) that are required by such algorithms. When this study focuses on the resources that are necessary for any algorithm that solves a particular task (or a task of a particular type), it is viewed as belonging to the theory of Computational Complexity (also known as Complexity Theory). In contrast, when the focus is on the design and analysis of specific algorithms (rather than on the intrinsic complexity of the task), the study is viewed as belonging to a related area that may be called Algorithmic Design and Analysis. Furthermore, Algorithmic Design and Analysis tends to be sub-divided according to the domain of mathematics, science, and engineering in which the computational tasks arise.