We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Usually, a proof of a theorem contains more knowledge than the mere fact that the theorem is true. For instance, to prove that a graph is Hamiltonian it suffices to exhibit a Hamiltonian tour in it; however, this seems to contain more knowledge than the single bit Hamiltonian/non-Hamiltonian.In this paper a computational complexity theory of the “knowledge” contained in a proof is developed. Zero-knowledge proofs are defined as those proofs that convey no additional knowledge other than the correctness of the proposition in question. Examples of zero-knowledge proof systems are given for the languages of quadratic residuosity and 'quadratic nonresiduosity. These are the first examples of zero-knowledge proofs for languages not known to be efficiently recognizable.

/pdf/the-knowledge-complexity-of-interactive-proof-systems-31apre0ecf.pdf

The knowledge complexity of interactive proof-systems

PART I: FOUNDATIONS 1. Introduction to Fixed-Parameter Algorithms 2. Preliminaries and Agreements 3. Parameterized Complexity Theory - A Primer 4. Vertex Cover - An Illustrative Example 5. The Art of Problem Parameterization 6. Summary and Concluding Remarks PART II: ALGORITHMIC METHODS 7. Data Reduction and Problem Kernels 8. Depth-Bounded Search Trees 9. Dynamic Programming 10. Tree Decompositions of Graphs 11. Further Advanced Techniques 12. Summary and Concluding Remarks PART III: SOME THEORY, SOME CASE STUDIES 13. Parameterized Complexity Theory 14. Connections to Approximation Algorithms 15. Selected Case Studies 16. Zukunftsmusik References Index

Invitation to fixed-parameter algorithms

Disjunctive Logic Programming (DLP) is an advanced formalism for knowledge representation and reasoning, which is very expressive in a precise mathematical sense: it allows one to express every property of finite structures that is decidable in the complexity class ΣP2 (NPNP). Thus, under widely believed assumptions, DLP is strictly more expressive than normal (disjunction-free) logic programming, whose expressiveness is limited to properties decidable in NP. Importantly, apart from enlarging the class of applications which can be encoded in the language, disjunction often allows for representing problems of lower complexity in a simpler and more natural fashion.This article presents the DLV system, which is widely considered the state-of-the-art implementation of disjunctive logic programming, and addresses several aspects. As for problem solving, we provide a formal definition of its kernel language, function-free disjunctive logic programs (also known as disjunctive datalog), extended by weak constraints, which are a powerful tool to express optimization problems. We then illustrate the usage of DLV as a tool for knowledge representation and reasoning, describing a new declarative programming methodology which allows one to encode complex problems (up to ΔP3-complete problems) in a declarative fashion. On the foundational side, we provide a detailed analysis of the computational complexity of the language of DLV, and by deriving new complexity results we chart a complete picture of the complexity of this language and important fragments thereof.Furthermore, we illustrate the general architecture of the DLV system, which has been influenced by these results. As for applications, we overview application front-ends which have been developed on top of DLV to solve specific knowledge representation tasks, and we briefly describe the main international projects investigating the potential of the system for industrial exploitation. Finally, we report about thorough experimentation and benchmarking, which has been carried out to assess the efficiency of the system. The experimental results confirm the solidity of DLV and highlight its potential for emerging application areas like knowledge management and information integration.

The DLV system for knowledge representation and reasoning

Supervised classification is one of the tasks most frequently carried out by so-called Intelligent Systems. Thus, a large number of techniques have been developed based on Artificial Intelligence (Logic-based techniques, Perceptron-based techniques) and Statistics (Bayesian Networks, Instance-based techniques). The goal of supervised learning is to build a concise model of the distribution of class labels in terms of predictor features. The resulting classifier is then used to assign class labels to the testing instances where the values of the predictor features are known, but the value of the class label is unknown. This paper describes various classification algorithms and the recent attempt for improving classification accuracy--ensembles of classifiers.

/pdf/machine-learning-a-review-of-classification-and-combining-1633dmcgns.pdf

Machine learning: a review of classification and combining techniques

This article surveys various complexity and expressiveness results on different forms of logic programming. The main focus is on decidable forms of logic programming, in particular, propositional logic programming and datalog, but we also mention general logic programming with function symbols. Next to classical results on plain logic programming (pure Horn clause programs), more recent results on various important extensions of logic programming are surveyed. These include logic programming with different forms of negation, disjunctive logic programming, logic programming with equality, and constraint logic programming.

/pdf/complexity-and-expressive-power-of-logic-programming-334cjkp9dg.pdf

Complexity and expressive power of logic programming

This paper surveys various complexity results on different forms of logic programming. The main focus is on decidable forms of logic programming, in particular propositional logic programming and datalog, but we also mention general logic programming with function symbols. Next to classical results on plain logic programming (pure Horn clause programs), more recent results on various important extensions of logic programming are surveyed. These include logic programming with different forms of negation, disjunctive logic programming, logic programming with equality, and constraint logic programming. The complexity of the unification problem is also addressed.

http://www.math.ucsd.edu/~sbuss/CourseWeb/Math268_2007WS/DantsinEtal02TCS.pdf

A deterministic (2 - 2/( k + 1)) n algorithm for k -SAT based on local search

There are many algorithms for testing satisfiability — how to evaluate and compare them? It is common (but still disputable) to identify the efficiency of an algorithm with its worst-case complexity. From this point of view, asymptotic upper bounds on the worst-case running time and space is a criterion for evaluation and comparison of algorithms. In this chapter we survey ideas and techniques behind satisfiability algorithms with the currently best upper bounds. We also discuss some related questions: “easy” and “hard” cases of SAT, reducibility between various restricted cases of SAT, the possibility of solving SAT in subexponential time, etc. In Section 12.1 we define terminology and notation used throughout the chapter. Section 12.2 addresses the question of which special cases of SAT are polynomial-time tractable and which ones remain NP-complete. The first nontrivial upper bounds for testing satisfiability were obtained for algorithms that solve k-SAT; such algorithms also form the core of general SAT algorithms. Section 12.3 surveys the currently fastest algorithms for k-SAT. Section 12.4 shows how to use bounds for k-SAT to obtain the currently best bounds for SAT. Section 12.5 addresses structural questions like “what else happens if k-SAT is solvable in time 〈. . .〉?”. Finally, Section 12.6 summarizes the currently best bounds for the main cases of the satisfiability problem.

Worst-Case Upper Bounds

The aim of this paper is to generalize logic programs, for dealing with probabilistic knowledge. Using the possible-worlds approach of probabilistic logic ([Nil]), we define probabilistic logic programs so that their clauses may be true or false with some probabilities and goals may succeed or fail with probabilities too. Probabilistic logic programs may contain negation, their semantics agrees with negation as failure (unlike probabilistic logic which is based on the standard logical negation).

Evgeny Dantsin

Papers

Complexity and expressive power of logic programming

Complexity and expressive power of logic programming

A deterministic (2 - 2/( k + 1)) n algorithm for k -SAT based on local search

Worst-Case Upper Bounds

Probabilistic logic programs and their semantics