Shorter tours by nicer ears: 7/5-approximation for the graph-TSP, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs
We prove new results for approximating the graph-TSP and some related problems. We obtain polynomial-time algorithms with improved approximation guarantees.
For the graph-TSP itself, we improve the approximation ratio to 7/5. For a generalization, the minimum T-tour problem, we obtain the first nontrivial approximation algorithm, with ratio 3/2. This contains the s-t-path graph-TSP as a special case. Our approximation guarantee for finding a smallest 2-edge-connected spanning subgraph is 4/3.
The key new ingredient of all our algorithms is a special kind of ear-decomposition optimized using forest representations of hypergraphs. The same methods also provide the lower bounds (arising from LP relaxations) that we use to deduce the approximation ratios.
Mathematics Subject Classification (2000)90C27 05C85 68R10
Unable to display preview. Download preview PDF.
- A. Alexander, S. Boyd, and P. Elliott-Magwood: On the integrality gap of the 2-edge connected subgraph problem, Technical Report TR-2006-04, SITE, University of Ottawa, 2006.Google Scholar
- H.-C. An, R. Kleinberg, and D. B. Shmoys: Improving Christofides’ algorithm for the s-t path TSP, Proceedings of the 44th Annual ACM Symposium on Theory of Computing (2012), 875–886Google Scholar
- S. Boyd, R. Sitters, S. van der Ster and L. Stougie: TSP on cubic and subcubic graphs, in: Integer Programming and Combinatorial Optimization; Proceedings of the 15th IPCO Conference; LNCS 6655 (O. Günlük, G.J. Woeginger, eds.), Springer Berlin 2011, 65–77.Google Scholar
- 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
- W. J. Cook: In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation, Princeton University Press 2012.Google Scholar
- J. Edmonds: The Chinese postman’s problem, Bulletin of the Oper-ations Research Society of America 13 (1965), 373.Google Scholar
- J. Edmonds: Submodular functions, matroids and certain polyhedra, in: Combinatorial Structures and Their Applications; Proceedings of the Calgary International Conference on Combinatorial Structures and Their Applications 1969 (R. Guy, H. Hanani, N. Sauer, J. Schönheim, eds.), Gordon and Breach New York 1970, 69–87.Google Scholar
- H. N. Gabow: Implementation of algorithms for maximum matching on nonbipartite graphs, Ph.D. thesis, Department of Computer Science, Stanford University 1973.Google Scholar
- T. Mömke and O. Svensson: Approximating graphic TSP by matchings, Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science (2011), 560–569.Google Scholar
- M. Mucha: 13/9-approximation for graphic TSP, Proceedings of the 29th Symposium on Theoretical Aspects of Computer Science (2012), 30–41.Google Scholar
- S. Oveis Gharan, A. Saberi and M. Singh: A randomized rounding approach to the traveling salesman problem, Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science (2011), 550–559.Google Scholar
- J. Vygen: New approximation algorithms for the TSP, OPTIMA 90 (2012), 1–12.Google Scholar