SC-Hamiltonicity and Its Linkages with Strong Hamiltonicity of a Graph
SIAM Journal on Discrete Mathematics
An O(n4) Algorithm for the QAP Linearization Problem
Mathematics of Operations Research
Three value TSP and linkages with the three value linear spanning 2-forests
Discrete Applied Mathematics
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A polynomial time testable characterization of cost matrices associated with a complete digraph on $n$ nodes such that all the Hamiltonian cycles (tours) have the same cost is well known. Tarasov [U.S.S.R. Comput. Maths. Math. Phys., 21 (1981), pp. 167-174.] obtained a characterization of cost matrices where tour costs take two distinct values. We provide a simple alternative characterization of such cost matrices, which can be tested in $O(n^2)$ time. We also provide analogous results where tours are replaced by Hamiltonian paths. When the cost matrix is skew-symmetric, we provide polynomial time testable characterizations such that the tour costs take three distinct values. Corresponding results for the case of Hamiltonian paths are also given. Using these results, special instances of the asymmetric traveling salesman problem (ATSP) are identified that are solvable in polynomial time and that have improved constant factor approximation schemes. In particular, we observe that the 3/2 performance guarantee of the Christofides algorithm extends to all metric Hamiltonian symmetric matrices. Further, we identify special classes of ATSP for which polynomial $\epsilon$-approximation algorithms are available for $\epsilon \in \{3/2, 4/3, 4\tau, \frac{3\tau^2}{2}, \frac{4+\delta}{3}\}$, where $\tau 1/2$ and $\delta \geq 0$ are constants.