The paper introduces three extensions of the A* search algorithm which improve the search efficiency by relaxing the admissibility condition. 1) A* employs an admissible heuristic function but invokes quicker termination conditions while still guaranteeing that the cost of the solution found will not exceed the optimal cost by a factor greater than 1 + . 2) RÂ¿* may employ heuristic functions which occasionally violate the admissibility condition, but guarantees that at termination the risk of missing the opportunity for further cost reduction is at most Â¿. 3) RÂ¿*,* is a speedup version of RÂ¿*, combining the termination condition of A* with the risk-admissibility condition of RÂ¿*. The Traveling Salesman problem was used as a test vehicle to examine the performances of the algorithms A* and RÂ¿*. The advantages of A* are shown to be significant in difficult problems, i.e., problems requiring a large number of expansions due to the presence of many subtours of roughly equal costs. The use of RÂ¿* is shown to produce a 4:1 reduction in search time with only a minor increase in final solution cost.