A note on the prize collecting traveling salesman problem
We study the version of the prize collecting traveling salesman problem, where the objective is to find a tour that visits a subset of vertices such that the length of the tour plus the sum of penalties associated with vertices not in the tour is as small as possible. We present an approximation algorithm with constant bound. The algorithm is based on Christofides' algorithm for the traveling salesman problem as well as a method to round fractional solutions of a linear programming relaxation to integers, feasible for the original problem.
Key wordsLinear programming prize collecting rounding fractional solutions traveling salesman problem worst-case analysis
Unable to display preview. Download preview PDF.
- E. Balas, “The prize collecting traveling salesman problem,”Networks 19 (1989) 621–636.Google Scholar
- N. Christofides, “Worst-case analysis of a new heuristic for the traveling salesman problem,” Report 388, Graduate School of Industrial Administration, Carnegie-Mellon University (Pittsburgh, PA, 1976).Google Scholar
- M.X. Goemans and D.J. Bertsimas, “On the parsimonious property of connectivity problems,” in:Proceeding of the 1st ACM-SIAM Symposium on Discrete Algorithms (San Francisco, CA, 1990).Google Scholar
- M. Held and R.M. Karp, “The traveling salesman problem and minimum spanning trees,”Operations Research 18 (1970) 1138–1162.Google Scholar
- D.S. Johnson and C.H. Papadimitriou, “Performance guarantees for heuristics,” in: E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan and D.B. Shmoys, eds.,The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization (Wiley, New York, 1985) pp. 145–180.Google Scholar
- J.K. Lenstra, D.B. Shmoys and E. Tardos, “Approximation algorithms for scheduling unrelated parallel machines,” in:Proceedings of the 28th Annual IEEE Symposium on the Foundation of Computer Science (Computer Society Press of the IEEE, New York, 1987) pp. 217–224.Google Scholar
- L. Lovasz, “On some connectivity properties of Eulerian multigraphs,”Acta Mathematica Academiae Scientiarum Hungaricae 28 (1976) 129–138.Google Scholar
- L. Lovasz,Combinatorial Problems and Exercises (North-Holland, Amsterdam, 1979).Google Scholar
- D. Shmoys and D. Williamson, “Analyzing the Held—Karp TSP bound: A monotonicity property with application,” to appear in:Information Processing Letters (1988).Google Scholar
- L. Wolsey, “Heuristic analysis, linear programming and branch and bound,”Mathematical Programming Study 13 (1980) 121–134.Google Scholar