P-Complete Approximation Problems
Journal of the ACM (JACM)
Confronting hardness using a hybrid approach
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
On The Approximability Of The Traveling Salesman Problem
Combinatorica
Reoptimization of the metric deadline TSP
Journal of Discrete Algorithms
On the hardness of reoptimization
SOFSEM'08 Proceedings of the 34th conference on Current trends in theory and practice of computer science
Algorithmics for Hard Problems: Introduction to Combinatorial Optimization, Randomization, Approximation, and Heuristics
Paired approximation problems and incompatible inapproximabilities
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Reoptimization of minimum and maximum traveling salesman's tours
SWAT'06 Proceedings of the 10th Scandinavian conference on Algorithm Theory
Improved approximations for TSP with simple precedence constraints
CIAC'10 Proceedings of the 7th international conference on Algorithms and Complexity
Parameterized Complexity
On the approximation ratio of the path matching christofides algorithm
SOFSEM'12 Proceedings of the 38th international conference on Current Trends in Theory and Practice of Computer Science
| Hi-index | 0.00 |
The metric traveling salesman problem is one of the most prominent APX-complete optimization problems. An important particularity of this problem is that there is a large gap between the known upper bound and lower bound on the approximability (assuming P ≠ NP). In fact, despite more than 30 years of research, no one could find a better approximation algorithm than the 1.5-approximation provided by Christofides. The situation is similar for a related problem, the metric Hamiltonian path problem, where the start and the end of the path are prespecified: the best approximation ratio up to date is 5/3 by an algorithm presented by Hoogeveen almost 20 years ago. In this paper, we provide a tight analysis of the combined outcome of both algorithms. This analysis reveals that the sets of the hardest input instances of both problems are disjoint in the sense that any input is guaranteed to allow at least one of the two algorithms to achieve a significantly improved approximation ratio. In particular, we show that any input instance that leads to a 5/3-approximation with Hoogeveen's algorithm enables us to find an optimal solution for the traveling salesman problem. This way, we determine a set S of possible pairs of approximation ratios. Furthermore, for any input we can identify one pair of approximation ratios within S that forms an upper bound on the achieved approximation ratios.