Solvable Case Of Traveling Salesman Problem Research Articles

A comparison of lower bounds for the symmetric circulant traveling salesman problem

When the matrix of distances between cities is symmetric and circulant, the traveling salesman problem (TSP) reduces to the so-called symmetric circulant traveling salesman problem (SCTSP), that has applications in the design of reconfigurable networks, and in minimizing wallpaper waste. The complexity of the SCTSP is open, but conjectured to be NP-hard, and we compare different lower bounds on the optimal value that may be computed in polynomial time. We derive a new linear programming (LP) relaxation of the SCTSP from the semidefinite programming (SDP) relaxation in [E. de Klerk, D.V. Pasechnik, R. Sotirov, On semidefinite programming relaxation of the traveling salesman problem, SIAM Journal of Optimization 19 (4) (2008) 1559–1573]. Further, we discuss theoretical and empirical comparisons between this new bound and three well-known bounds from the literature, namely the Held-Karp bound [M. Held, R.M. Karp, The traveling salesman problem and minimum spanning trees, Operations Research 18 (1970) 1138–1162], the 1-tree bound, and the closed-form bound for SCTSP proposed in [J.A.A. van der Veen, Solvable cases of TSP with various objective functions, Ph.D. Thesis, Groningen University, The Netherlands, 1992].

Pyramidal traveling salesman problem

In this paper, we give new polynomially testable sufficiency conditions for a given instance of the traveling salesman problem (TSP) to have an optimal tour that is pyramidal. This properly generalizes the Demidenko condition and the conditions of Warren. We thus have new, nontrivial polynomially testable and polynomially solvable cases of TSP. Scope and purpose The problem of identifying polynomially testable and polynomially solvable subclasses of the TSP is of significant theoretical and practical value. Pyramidal TSP is a well-studied subclass of the TSP that is polynomially solvable. However, it is an NP-hard problem to test if a given instance of TSP is pyramidal. Various polynomially testable subclasses of this have been identified in literature. In this paper, we give a general polynomially testable sufficiency condition for an instance of TSP to be pyramidal. This provides simpler proofs and also a proper generalization of the known classes.

Gilmore–Gomory type traveling salesman problems

One of the well-known, solvable cases of the traveling salesman problem (TSP) is the Gilmore–Gomory case. A generalization of their algorithm to what we call the GG scheme is known to solve a fairly large subclass of the problem. In this paper, we identify new classes of TSP for which the GG scheme produces an optimal solution and which properly include the class of TSP for which the cost matrix is a permuted distribution matrix. Implementation of the GG scheme is NP-hard for these classes of TSP. However, we identify some subclasses for which the GG scheme can be implemented in polynomial time. Some of these new solvable cases are proper generalizations of some well-known cases. Scope and purpose Since TSP is NP-hard, it is interesting to identify, generalize and unify polynomially testable and polynomially solvable subclasses of the TSP. Perhaps, the most well-known solvable case of TSP is the one identified by Gilmore and Gomory in 1964. We show that their approach can be generalized and applied to a fairly larger class of TSP. We point out some polynomially testable classes of TSP for which the approach can be implemented in polynomial time. Some of these new solvable cases generalize and unify some well-known cases.

A generalization of the convex-hull-and-line traveling salesman problem

Two instances of the traveling salesman problem, on the same node set {1,2,…,n} but with different cost matrices C and C′ , are equivalent iff there exist {ai,bi: i=1,…, n} such that for any 1≤i, j≤n,i≠j,C′(i,j)=C(i,j)+ai+bj [7]. One of the well-solved special cases of the traveling salesman problem (TSP) is the convex-hull-and-line TSP. We extend the solution scheme for this class of TSP given in [9] to a more general class which is closed with respect to the above equivalence relation. The cost matrix in our general class is a certain composition of Kalmanson matrices. This gives a new, non-trivial solvable case of TSP.