Thisvolumecontainsthepaperspresentedatthe7th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX 2004) and the 8th International Workshop on Randomization and Compu- tion (RANDOM 2004), which took place concurrently at Harvard University, Cambridge, on August 22-24, 2004. APPROX focuses on algorithmic and c- plexity issues surrounding the development of e?cient approximate solutions to computationally hard problems, and this year's workshop was the seventh in the series after Aalborg (1998), Berkeley (1999), Saarbru ]cken (2000), Berkeley (2001), Rome (2002), and Princeton (2003). RANDOM is concerned with app- cations of randomness to computational and combinatorial problems, and this year'sworkshopwasthe eighth in the seriesfollowing Bologna(1997), Barcelona (1998), Berkeley (1999), Geneva (2000), Berkeley (2001), Harvard (2002), and Princeton (2003). Topics of interest for APPROX and RANDOM are: design and analysis of approximation algorithms, inapproximability results, approximationclasses, - line problems, small space and data streaming algorithms, sub-linear time al- rithms, embeddings and metric space methods in approximation, math prog- ming in approximation algorithms, coloring and partitioning, cuts and conn- tivity, geometric problems, network design and routing, packing and covering, scheduling, game theory, design and analysis of randomized algorithms, r- domized complexity theory, pseudorandomness and derandomization, random combinatorial structures, random walks/Markov chains, expander graphs and randomness extractors, probabilistic proof systems, random projectionsand - beddings, error-correctingcodes, average-caseanalysis, propertytesting, com- tational learning theory, and other applications of approximation and rand- ness. The volumecontains19+18contributed papers, selected by the two program committees from 54+33 submissions received in response to the call for papers."

This volume contains a collection of studies in the areas of complexity theory and property testing. The 21 pieces of scientific work included were conducted at different times, mostly during the last decade. Although most of these works have been cited in the literature, none of them was formally published before. Within complexity theory the topics include constant-depth Boolean circuits, explicit construction of expander graphs, interactive proof systems, monotone formulae for majority, probabilistically checkable proofs (PCPs), pseudorandomness, worst-case to average-case reductions, and zero-knowledge proofs. Within property testing the topics include distribution testing, linearity testing, lower bounds on the query complexity (of property testing), testing graph properties, and tolerant testing. A common theme in this collection is the interplay between randomness and computation.

Property testing algorithms are ultra-efficient algorithms that decide whether a given object (e.g., a graph) has a certain property (e.g., bipartiteness), or is significantly different from any object that has the property. To this end property testing algorithms are given the ability to perform (local) queries to the input, though the decisions they need to make usually concern properties with a global nature. In the last two decades, property testing algorithms have been designed for many types of objects and properties, amongst them, graph properties, algebraic properties, geometric properties, and more. In this book the authors survey results in property testing, with an emphasis on common analysis and algorithmic techniques. Among the techniques surveyed are the following: a) The self-correcting approach, which was mainly applied in the study of property testing of algebraic properties. b) The enforce and test approach, which was applied quite extensively in the analysis of algorithms for testing graph properties (in the dense-graphs model), as well as in other contexts. c) Szemeredi's Regularity Lemma, which plays a very important role in the analysis of algorithms for testing graph properties (in the dense-graphs model). d) The approach of Testing by implicit learning, which implies efficient testability of membership in many functions classes. e) Algorithmic techniques for testing properties of sparse graphs, which include local search and random walks.