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.