The Symmetric Circulant Traveling Salesman Problem

Abstract

An n × n matrix D = d[i, j] is said to be circulant, if the entries d[i, j] verifying (j − i) = k mod n, for some k, have the same value (for a survey on circulant matrix properties, see Davis (1979)). A directed (respectively, undirected) graph is circulant, if its adjacency matrix is circulant (respectively, symmetric, and circulant). Similarly, a weighted graph is circulant, if its weighted adjacency matrix is circulant. In the last years, it had been often investigated if a graph problem becomes easier when it is restricted to the circulant graphs. For example, the Maximum Clique problem, and the Minimum Graph Coloring problem remain NP-hard, and not approximable within a constant factor, when the general instance is forced to be a circulant undirected graphs, as shown by Codenotti, et al. (1998). On the other hand, Muzychuk (2004) has proved that the Graph Isomorphism problem restricted to circulant undirected graphs is in P, while the general case is, probably, harder. It is still an open question whether the Directed Hamiltonian Circuit problem, restricted to circulant (directed) graphs, remains NP-hard, or not. A solution in some special cases has been found by Garfinkel (1977), Fan Yang, et al. (1997), and Bogdanowicz (2005). The Hamiltonian Circuit problem admits, instead, a polynomial time algorithm on the circulant undirected graphs, as shown by Burkard, and Sandholzer (1991). It leads to a polynomial time algorithm for the Bottleneck Traveling Salesman Problem on the symmetric circulant matrices. Finally, in Gilmore, et al. (1985) it is shown that the Shortest Hamiltonian Path problem is polynomial time solvable on the circulant matrices, while the general case is NP-hard. The positive results contained in Burkard, and Sandholzer (1991), and in Gilmore, et al. (1985) have encouraged the research on the Symmetric Circulant Traveling Salesman problem, that is, the Sum Traveling Salesman Problem restricted to the symmetric, and circulant matrices. In this chapter we deal with such problem, called for short SCTSP. In §1–§3 the problem is introduced, and the notation is fixed. In §4–§6 an overview is given on the last 16 year results. Firstly, an upper bound (§4.1), a lower bound (§4.2), and a polynomial time 2approximation algorithm for the general case of SCTSP (§4.3) are discussed. No better result concerning the computational complexity of SCTSP is known. Secondly, some sufficient theorems solving particular cases of SCTSP are presented (§5). Finally, §6 is devoted to a recently introduced subcase of SCTSP. §7 completes the chapter by presenting open problems, remarks, and future developments. O pe n A cc es s D at ab as e w w w .ite ch on lin e. co m