The paper focuses on finding the m best solutions to a combinatorial optimization problems using Best-First or Branch-and-Bound search. We are interested in graphical model optimization tasks (e.g., Weighted CSP), which can be formulated as finding the m-best solution-paths in a weighted search graph. Specifically, we present m-A*, extending the well-known A* to the m-best problem, and prove that all A*’s properties are maintained, including soundness and completeness of m- A*, dominance with respect to improved heuristics and most significantly optimal efficiency compared with any other search algorithm that use the same heuristic function. We also present and analyse m-B&B, an extension of a Depth First Branch and Bound algorithm to the task of finding the m best solutions. Finally, for graphical models, a hybrid of A* and a variable-elimination scheme yields an algorithm which has the best complexity bound compared with earlier known m-best algorithms.