Greedy Local Search

Greedy local search search methods have a long history in the area of com-binatorial optimization. One of the earliest application of local search was to nd good solutions for the Traveling Salesman Problem (TSP). In this problem, one is given a weighted graph and the goal is to nd the shortest closed path that visits each node exactly once. The TSP is prototypical of the large class of NP-hard optimization problems, for which it is widely believed that no eecient (polynomial time) algorithm exists (Cook 1971; Garey and Johnson 1979; Papadimitriou and Steiglitz 1982). A local search method for the TSP proceeds as follows. Start with a randomly generated closed path. Subsequently, we make small (\local") changes to the path to try to nd a shorter one. One example of such a local change is the so-called k?change, in which we remove k edges from the path and replace them with k other edges. A k?change is made whenever it improves the current solution. If no more local improvement can be found, the procedure terminates. Lin (1965) and Lin and Kernighan (1973) showed that this simple procedure, using k = 3, leads to solutions that are surprisingly close to the optimal solution. The basic local search framework allows for several variations. For example , there is the choice of the initial solution, the nature of the local changes considered, and the manner in which the actual improvement of the current solution is selected. Lin and Kernighan found that multiple runs with diierent random initial paths would lead to the best solutions. Somewhat surprisingly, starting with \good" initial paths did not necessarily lead to better nal solutions. The reason for this appears to be that the local search mechanism itself is powerful enough to improve upon the initial solutions | often quickly giving better solutions than those generated using 1