Well-solvable special cases of the TSP : a survey

The Traveling Salesman Problem belongs to the most important and most investigated problems in combinatorial optimization. Although it is an NP-hard problem, many of its special cases can be solved efficiently. We survey these special cases with emphasis on results obtained during the decade 1985-1995. This survey complements an earlier survey from 1985 compiled by Gilmore, Lawler and Shmoys. Keywords: Traveling Salesman Problem, Combinatorial optimization, Polynomial time algorithm, Computational complexity

Traveling Salesman Problem

The idea behind TSP was conceived by Austrian mathematician Karl Menger in mid 1930s who invited the research community to consider a problem from the everyday life from a mathematical point of view. A traveling salesman has to visit exactly once each one of a list of m cities and then return to the home city. He knows the cost of traveling from any city i to any other city j. Thus, which is the tour of least possible cost the salesman can take? In this book the problem of finding algorithmic technique leading to good/optimal solutions for TSP (or for some other strictly related problems) is considered. TSP is a very attractive problem for the research community because it arises as a natural subproblem in many applications concerning the every day life. Indeed, each application, in which an optimal ordering of a number of items has to be chosen in a way that the total cost of a solution is determined by adding up the costs arising from two successively items, can be modelled as a TSP instance. Thus, studying TSP can never be considered as an abstract research with no real importance

Hamiltonian cycles in circulant digraphs with two stripes

The circulant traveling salesman problem (CTSP) is the problem of finding a minimum weight Hamiltonian cycle in a weighted graph with circulant distance matrix. The computational complexity of this problem is not known. In fact, even the complexity of deciding Hamiltonicity of the underlying graph is unknown. This paper provides a characterization of Hamiltonian digraphs with circulant distance matrix containing only two nonzero stripes. The corresponding conditions can be checked in polynomial time. Secondly, we show that all Hamiltonian cycles of a circulant 2-digraph are periodic. Based on these two results, a method for enumerating all Hamiltonian cycles in such digraphs is described. Moreover, two simple algorithms are derived for solving the sum and bottleneck versions of CTSP for circulant distance matrices with two nonzero stripes