Heuristics for sparse general travelling salesman problems

The well-known travelling salesman problem can be expressed in graphtheoretic terms as follows : find a minimum-weight Hamiltonian cycle in a connected edge-weighted graph G(N, A). vVe concentrate here on the problem where G is large and sparse, i.e., where INI is 500 or more and IAI is O( INI). Such graphs arise frequently in practice, where G represents a road network, and each node (city) is connected directly to 3 or 4 other nodes. Sparse graphs may not have Hamiltonian cycles and hence there may be no solution to the problem as stated. vVhether this is the case or not, the requirement that the cycle is Hamiltonian is often unnecessarily restrictive. It is sufficient in many practical problems, such as traffic planning. to address only what is called the General Travelling Salesman Problem, which requires merely a minimum-weight, closed spanning walk, in which any node can be repeated if this lowers the total weight. Such a walk can, of course, be found in any connected graph. vVe report on the results of an experimental comparison of some heuristic methods for this problem.