We present a new polynomial-space algorithm, called Adopt, for distributed constraint optimization (DCOP). DCOP is able to model a large class of collaboration problems in multi-agent systems where a solution within given quality parameters must be found. Existing methods for DCOP are not able to provide theoretical guarantees on global solution quality while operating both efficiently and asynchronously. Adopt is guaranteed to find an optimal solution, or a solution within a user-specified distance from the optimal, while allowing agents to execute asynchronously and in parallel. Adopt obtains these properties via a distributed search algorithm with several novel characteristics including the ability for each agent to make local decisions based on currently available information and without necessarily having global certainty. Theoretical analysis shows that Adopt provides provable quality guarantees, while experimental results show that Adopt is significantly more efficient tha...