Weighted search (best-first or depth-first) refers to search with a heuristic function multiplied by a constant w [Pohl (1970)]. The paper shows for the first time that for graphical models optimization queries weighted best-first and weighted depth-first Branch and Bound search schemes are competitive energy-minimization anytime optimization algorithms. Weighted best-first schemes were investigated for path-finding tasks, however, their potential for graphical models was ignored, possibly because of their memory costs and because the alternative depth-first Branch and Bound seemed very appropriate for bounded depth. The weighted depth-first search has not been studied for graphical models. We report on a significant empirical evaluation, demonstrating the potential of both weighted best-first search and weighted depth-first Branch and Bound algorithms as approximation anytime schemes (that have suboptimality bounds) and compare against one of the best depth-first Branch and Bound solvers to date.