Papadimitriou Complexity

[Mireille Kreines]

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This text offers a comprehensive and accessible treatment of the theory of algorithms and complexity - the elegant body of concepts and methods developed by computer scientists over the past 30 years for studying the performance and limitations of computer algorithms. Among topics covered are: reductions and NP-completeness, cryptography and protocols, randomized algorithms, and approximability of optimization problems, circuit complexity, the "structural" aspects of the P=NP question, parallel computation, the polynomial hierarchy, and many others. Several sophisticated and recent results are presented in a rather simple way, while many more are developed in the form of extensive notes, problems, and hints. The book is surprisingly self-contained, in that it develops all necessary mathematical prerequisites from such diverse fields as computability, logic, number theory, combinatorics and probability.

Copies of the classnotes are on the internet in PDF format as given below. The "Proofs of Theorems" files were prepared in Beamer. The "Printout of Proofs" are printable PDF files of the Beamer slides without the pauses. These notes and supplements have not been classroom tested (and so may have some typographical errors).

Computational Complxity is not a formal class at ETSU. These notes are meant for self-study and reference.Classes which cover similar topics are offered by the Department of Computer Science (these class descriptions are from the ETSU 2022-23 Graduate Catalog): Formal Languages and Computational Complexity (CSCI 5610). Prerequisites: MATH 2710, CSCI 2210 or consent of the instructor.Problem-solving is a fundamental aspect of computer science. This course teaches students how to reduce a computational problem to its simplest form and analyze the problem to determine its inherent computational complexity. Topics include formal languages and automata theory, Turing machines, computational complexity, and the theory of NP-completeness.When Offered: Variable. Analysis Of Algorithms (CSCI 5620).Prerequisites: Differential and integral calculus, discrete structures, data structures. This course covers basic techniques for analyzing algorithmic complexity. It describes the design and analysis of selected algorithms for solving important problems that arise often in applications of computer science, including sorting, selection, graph theory problems (e.g., shortest path, graph traversals), string matching, dynamic programming problems, NP-complete problems. When Offered: Fall, alternate years [odd falls].The first of these CSCI classes is most similar to the material presented here. However, Formal Languages and Computational Complexity (CSCI 5610) only has prerequisites of Discrete Structures (MATH 2710) or Data Structures (CSCI 2210). Discrete Structures (MATH 2710) was discontinued in the ealy 2000s, though I have online notes for Discrete Structures from the last time I taught it (spring 2001). For the notes presented here, prerequisite material would include Mathematical Reasoning (MATH 3000) and Introduction to Set Theory (not a formal ETSU class; some of this material is covered in Mathematical Reasoning, but not deeply). Some exposure to graph theory is desirable, since several of the initial problems encountered here involve graph theory problems. A sufficient background is given by Introduction to Graph Theory (MATH 4347/5347).

In computer science, PPAD ("Polynomial Parity Arguments on Directed graphs") is a complexity class introduced by Christos Papadimitriou in 1994. PPAD is a subclass of TFNP based on functions that can be shown to be total by a parity argument.[1][2] The class attracted significant attention in the field of algorithmic game theory because it contains the problem of computing a Nash equilibrium: this problem was shown to be complete for PPAD by Daskalakis, Goldberg and Papadimitriou with at least 3 players and later extended by Chen and Deng to 2 players.[3][4]

Subclasses of TFNP are defined based on the type of mathematical proof used to prove that a solution always exists. Informally, PPAD is the subclass of TFNP where the guarantee that there exists a y such that P(x,y) holds is based on a parity argument on a directed graph. The class is formally defined by specifying one of its complete problems, known as End-Of-The-Line:

PPAD is contained in (but not known to be equal to) PPA (the corresponding class of parity arguments for undirected graphs) which is contained in TFNP. PPAD is also contained in (but not known to be equal to) PPP, another subclass of TFNP. It contains CLS.[5]

PPAD is a class of problems that are believed to be hard, but obtaining PPAD-completeness is a weaker evidence of intractability than that of obtaining NP-completeness. PPAD problems cannot be NP-complete, for the technical reason that NP is a class of decision problems, but the answer of PPAD problems is always yes, as a solution is known to exist, even though it might be hard to find that solution.[6] However, PPAD and NP are closely related. While the question whether a Nash equilibrium exists for a given game cannot be NP-hard because the answer is always yes, the question whether a second equilibrium exists is NP complete.[7] Examples of PPAD-complete problems include finding Nash equilibria, computing fixed points in Brouwer functions, and finding Arrow-Debreu equilibria in markets.[8]

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Algorithms, perhaps together with Moore's law, compose the engine of the information technology revolution, whereas complexity--the antithesis of algorithms--is one of the deepest realms of mathematical investigation. After introducing the basic concepts of algorithms and complexity, and the fundamental complexity classes P (polynomial time) and NP (nondeterministic polynomial time, or search problems), we discuss briefly the P vs. NP problem. We then focus on certain classes between P and NP which capture important phenomena in the social and life sciences, namely the Nash equlibrium and other equilibria in economics and game theory, and certain processes in population genetics and evolution. Finally, an algorithm known as multiplicative weights update (MWU) provides an algorithmic interpretation of the evolution of allele frequencies in a population under sex and weak selection. All three of these equivalences are rife with domain-specific implications: The concept of Nash equilibrium may be less universal--and therefore less compelling--than has been presumed; selection on gene interactions may entail the maintenance of genetic variation for longer periods than selection on single alleles predicts; whereas MWU can be shown to maximize, for each gene, a convex combination of the gene's cumulative fitness in the population and the entropy of the allele distribution, an insight that may be pertinent to the maintenance of variation in evolution.

Vassiliki Voula Papadimitriou is a renowned computer scientist who has made significant contributions to the fields of theoretical computer science, algorithms, and complexity theory. She is a Professor of Computer Science at the University of California, Berkeley, and has received numerous awards for her work, including the MacArthur Fellowship and the Knuth Prize.

Papadimitriou's research has focused on the development of efficient algorithms for solving complex problems. She has made important contributions to the areas of approximation algorithms, online algorithms, and randomized algorithms. Her work has had a major impact on the design and analysis of algorithms for a wide range of applications, including scheduling, routing, and network optimization.

In addition to her research, Papadimitriou is also a gifted educator and has written several textbooks on algorithms and complexity theory. She is a Fellow of the Association for Computing Machinery (ACM) and a member of the National Academy of Sciences.

Vassiliki Voula Papadimitriou is a renowned computer scientist who has made significant contributions to the fields of theoretical computer science, algorithms, and complexity theory. Her work has focused on the development of efficient algorithms for solving complex problems, and she has made important contributions to the areas of approximation algorithms, online algorithms, and randomized algorithms.

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