The Distributed Constraint Optimization Problem (DCOP) is a promising approach for modeling distributed reasoning tasks that arise in multiagent systems. Unfortunately, existing methods for DCOP are not able to provide theoretical guarantees on global solution quality while allowing agents to operate asynchronously. We show how this failure can be remedied by allowing agents to make local decisions based on conservative cost estimates rather than relying on global certainty as previous approaches have done. This novel approach results in a polynomial-space algorithm for DCOP named Adopt that is guaranteed to find the globally optimal solution while allowing agents to execute asynchronously and in parallel. Detailed experimental results show that on benchmark problems Adopt obtains speedups of several orders of magnitude over other approaches. Adopt can also perform bounded-error approximation – it has the ability to quickly find approximate solutions and, unlike heuristic search methods, still maintain a theoretical guarantee on solution quality.