Performance Guarantees for Approximation Algorithms Depending on Parametrized Triangle Inequalities
SIAM Journal on Discrete Mathematics
Performance Guarantees for the TSP with a Parameterized Triangle Inequality
WADS '99 Proceedings of the 6th International Workshop on Algorithms and Data Structures
Finding a hamiltonian cycle in the square of a block
Finding a hamiltonian cycle in the square of a block
On the approximation hardness of some generalizations of TSP
SWAT'06 Proceedings of the 10th Scandinavian conference on Algorithm Theory
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We consider the problem of finding a cheapest Hamiltonian path of a complete graph satisfying a relaxed triangle inequality, i.e., such that for some parameter β 1, the edge costs satisfy the inequality c({x,y}) ≤ β(c({x,z}) + c({z,y})) for every triple of vertices x, y, z. There are three variants of this problem, depending on the number of prespecified endpoints: zero, one, or two. For metric graphs, there exist approximation algorithms, with approximation ratio $\frac{3}{2}$ for the first two variants and $\frac{5}{3}$ for the latter one, respectively. Using results on the approximability of the Travelling Salesman Problem with input graphs satisfying the relaxed triangle inequality, we obtain for our problem approximation algorithms with ratio min($\beta^{2} + \beta,\frac{3}{2}\beta^{2})$ for zero or one prespecified endpoints, and $\frac{5}{3}\beta^{2}$ for two endpoints.