This book constitutes the joint refereed proceedings of the 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006 and the 10th International Workshop on Randomization and Computation, RANDOM 2006, held in Barcelona, Spain, in August 2006. The 44 revised full papers presented were carefully reviewed and selected from 105 submissions. Among the topics covered are design and analysis of approximation algorithms, hardness of approximation problems, small spaces and data streaming algorithms, sub-linear time algorithms, embeddings and metric space methods, mathematical programming methods, coloring and partitioning, cuts and connectivity, game theory, network design and routing, packing and covering, scheduling, design and analysis of randomized algorithms, randomized complexity theory, pseudorandomness, derandomization, random combinatorial structures, Markov chains, prohabalistic proof systems, error-correcting codes, etc.