The Efficiency of Algorithms

S uppose you were asked to plan an itinerary for a traveling salesman who must visit a number of cities . You are given a map on which the dis­ tances between the cities are marked and you are asked to find the shortest route that passes through all the cities and returns to the starting point. An ap­ proach to this problem that is certain to give the correct answer is to trace all the possible routes. measure their length and pick the shortest one. If the tour included more than a few cities. how­ ever. hundreds or thousands of routes would have to be checked. If there were 100 cities. then even the fastest comput­ ers would require weeks of calculation to find the shortest path. In the search for a quicker solution you might try some less rigorous meth­ ods. One idea that seems reasonable is always to visit nearby cities before going on to those farther away . You would soon discover. however. that this proce­ dure does not invariably yield the cor­ rect answer. Other shortcuts also fail. In fact. the best methods known for solving the problem are not much better than the obvious but laborious procedure of checking all the possible itineraries. M athematicians now suspect that this problem and many others like it may forever remain beyond our ability to solve in any efficient way. That specula­ tion itself. however. is unconfirmed; al­ though no faster methods of solution have been found. neither has it been proved that faster methods do not exist.