On patching algorithms for random asymmetric traveling salesman problems
Mathematical Programming: Series A and B
When is the Assignment Bound Tight for the Asymmetric Traveling-Salesman Problem?
SIAM Journal on Computing
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
Information Processing Letters
Approximation algorithms for the TSP with sharpened triangle inequality
Information Processing Letters
The probabilistic relationship between the assignment and asymmetric traveling salesman problems
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Computers and Intractability; A Guide to the Theory of NP-Completeness
Computers and Intractability; A Guide to the Theory of NP-Completeness
A new approximation algorithm for the asymmetric TSP with triangle inequality
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Approximation algorithms for asymmetric TSP by decomposing directed regular multigraphs
Journal of the ACM (JACM)
An improved approximation algorithm for the asymmetric TSP with strengthened triangle inequality
Journal of Discrete Algorithms
Minimum-weight cycle covers and their approximability
Discrete Applied Mathematics
An improved approximation algorithm for the ATSP with parameterized triangle inequality
Journal of Algorithms
Two Approximation Algorithms for ATSP with Strengthened Triangle Inequality
WADS '09 Proceedings of the 11th International Symposium on Algorithms and Data Structures
Minimum-weight cycle covers and their approximability
WG'07 Proceedings of the 33rd international conference on Graph-theoretic concepts in computer science
An improved approximation algorithm for the maximum TSP
Theoretical Computer Science
Deterministic algorithms for multi-criteria TSP
TAMC'11 Proceedings of the 8th annual conference on Theory and applications of models of computation
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In this paper, we study the relationship between the Asymmetric Traveling Salesman Problem (ATSP) and the Cycle Cover Problem in terms of the strength of the triangle inequality on the edge costs in the given complete directed graph instance, G=(V,E). The strength of the triangle inequality is captured by parametrizing the triangle inequality as follows. A complete directed graph G=(V,E) with a cost function c:E-R^+ is said to satisfy the @c-parametrized triangle inequality if @c(c(u,w)+c(w,v))=c(u,v) for all distinct u,v,w@?V. Then the graph G is called a @c-triangular graph. For any @c-triangular graph G, for @c=1, the ratio ATSP(G)AP(G) can become unbounded. The upper bound is shown constructively and can also be viewed as an approximation algorithm for ATSP with parametrized triangle inequality. We also consider the following problem: in a @c-triangular graph, does there exist a function f(@c) such that c"m"a"xc"m"i"n is bounded above by f(@c)? (Here c"m"a"x and c"m"i"n are the costs of the maximum cost and minimum cost edges respectively.) We show that when @c=13, no such function f(@c) exists.