An Improved Lower Bound on the Approximability of Metric TSP and Approximation Algorithms for the TSP with Sharpened Triangle Inequality
We essentially modify the method of Engebretsen [En99] in order to get a lower bound of 3813/3812 − ε on the polynomial-time approximability of the metric TSP for any ε > 0. This is an improvement over the lower bound of 5381/5380 − ε in [En99]. Using this approach we moreover prove a lower bound δβ on the approximability of Δβ-TSP for 1/2 < β < 1, where Δβ-TSP is a subproblem of the TSP whose input instances satisfy the β-sharpened triangle inequality cost(u; v) ≤ β· (cost(u; x) + cost(x; v)) for all vertices u; v; x.
We present three different methods for the design of polynomial-time approximation algorithms for Δβ-TSP with 1/2 < β < 1, where the approximation ratio lies between 1 and 3/2, depending on β.
KeywordsApproximation algorithms Traveling Salesman Problem
Unable to display preview. Download preview PDF.
- [Ar97]S. Arora: Nearly linear time approximation schemes for Euclidean TSP and other geometric problems. In: Proc. 38th IEEE FOCS, 1997, pp. 554–563.Google Scholar
- [BC99]M. A. Bender, C. Chekuri: Performance guarantees for the TSP with a parameterized triangle inequality. In: Proc. WADS’99, LNCS 1663, Springer 1999, pp. 80–85.Google Scholar
- [BHKSU99]H.-J. Böckenhauer, J. Hromkovič, R. Klasing, S. Seibert, W. Unger: Towards the Notion of Stability of Approximation Algorithms and the Traveling Salesman Problem. Extended abstract in: Proc. CIAC 2000, LNCS, to appear. Full version in: Electronic Colloquium on Computational Complexity, Report No. 31 (1999).Google Scholar
- [BK98]P. Berman, M. Karpinski: On some tighter inapproximability results. Technical Report TR98-029, Electronic Colloquium on Computational Complexity, 1998.Google Scholar
- [Chr76]N. Christofides: Worst-case analysis of a new heuristic for the traveling salesman problem. Technical Report 388, Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh, 1976.Google Scholar
- [En99]L. Engebretsen: An explicit lower bound for TSP with distances one and two. Extended abstract in: Proc. STACS’99, LNCS 1563, Springer 1999, pp. 373–382. Full version in: Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 46 (1999).Google Scholar
- [EJ70]J. Edmonds, E. L. Johnson: Matching: A Well-Solved Class of Integer Linear Programs. In: Proc. Calgary International Conference on Combinatorial Structures and Their Applications, Gordon and Breach 1970, pp. 89–92.Google Scholar
- [Ho96]D. S. Hochbaum (Ed.): Approximation Algorithms for NP-hard Problems. PWS Publishing Company 1996.Google Scholar
- [LLRS85]E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, D. B. Shmoys (Eds.): The Traveling Salesman Problem. John Wiley & Sons, 1985.Google Scholar
- [Mi96]I. S. B. Mitchell: Guillotine subdivisions approximate polygonal subdivisions: Part II — a simple polynomial-time approximation scheme for geometric k-MST, TSP and related problems. Technical Report, Dept. of Applied Mathematics and Statistics, Stony Brook 1996.Google Scholar