Computing and Software Science: State of the Art and Perspectives

Algorithmics, computation, optimization, complexity, combinatorics and knowledge representation are closely related sub-areas of Theoretical Computer Science. The following summary presents short descriptions of the twelve chapters in this topical part. This first part of the book covers a wide range of topics that may roughly be summarized under the words “computation” and “complexity”. The choice of topics reflects the massive developments in computer science over recent decades. Many research areas that used to be outside of traditional computer science have been conquered and attacked with computer science tools. For example, classical Euclidean geometry has led to the area computational geometry, classical graph theory has led to algorithmic graph theory, biology gave us the area of computational biology, from the social sciences we got computational social choice, and economics delivered the areas of algorithmic game theory and computational economics. Computer science has always been very successful in modelling communication systems, for example wireless networks. One of the biggest challenges in neuroscience consists in understanding how the human brain works and how the brain performs computations. The internet (as a gigantic decentralized computing system) has led to the areas data mining and knowledge harvesting. The first part of the book deals with some of these challenges and trends, and the following twelve chapters analyze some aspects of these developments. We present short descriptions of these chapters. The chapter “Some Estimated Likelihoods for Computational Complexity” by Williams [16] addresses some of the most fundamental open problems in computational complexity theory. Of course, we would like to know the answer to the P versus NP question (Is P=NP?). A slightly easier problem asks whether P=PSPACE (this statement should at least be easier to disprove than P=NP, as NP⊆PSPACE holds). Other central open questions in complexity theory concern the so-called Exponential Time Hypothesis (ETH) of Impagliazzo and Paturi [9], the Strong Exponential Time Hypothesis (SETH), and the Nondeterministic Strong Exponential Time Hypothesis (NSETH), which all form strengthenings of the statement P = NP. In his chapter, Ryan Williams states concrete probabilities with which he believes that the various open problems have positive answers: P = NP with probability 80%, ETH should hold with probability 70%, SETH with probability 25%, and so on. The body of the chapter deals with the reasons why and how Ryan arrived at these probabilities, with technical possibilities and impossibilities, and many other things. c © Springer Nature Switzerland AG 2019 B. Steffen and G. Woeginger (Eds.): Computing and Software Science, LNCS 10000, pp. 3–8, 2019. https://doi.org/10.1007/978-3-319-91908-9_1

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