Abstract Optimization in graphical models is an important problem which has been studied in many AI frameworks such as weighted CSP, maximum satisfifiability or probabilistic networks. By identifying conditionally independent subproblems, which are solved independently and whose optimum is cached, recent Branch and Bound algorithms offer better asymptotic time complexity. But the locality of bounds induced by decomposition often hampers the practical effects of this result because subproblems are often uselessly solved to optimality. Following the Russian Doll Search (RDS) algorithm, a possible approach to overcome this weakness is to (inductively) solve a relaxation of each subproblem to strengthen bounds. The algorithm obtained generalizes both RDS and treedecomposition based algorithms such as BTD or AND-OR Branch and Bound. We study its effifi- ciency on different problems, closing a very hard frequency assignment instance which has been open for more than 10 years