Abstract
We design and analyse approximation algorithms for the minimum-cost connected T-join problem: given an undirected graph G=(V,E) with nonnegative costs on the edges, and a set of nodes T⊆V, find (if it exists) a spanning connected subgraph H of minimum cost such that every node in T has odd degree and every node not in T has even degree; H may have multiple copies of any edge of G. Two well-known special cases are the TSP (T=∅) and the s,t path TSP (T={s,t}). Recently, An et al. (Proceedings of the 44th Annual ACM Symposium on Theory of Computing, pp. 875–886, 2012) improved on the long-standing \(\frac{5}{3}\) approximation guarantee for the latter problem and presented an algorithm based on LP rounding that achieves an approximation guarantee of \(\frac{1+\sqrt{5}}{2}\approx1.61803\).
We show that the methods of An et al. extend to the minimum-cost connected T-join problem. They presented a new proof for a \(\frac{5}{3}\) approximation guarantee for the s,t path TSP; their proof extends easily to the minimum-cost connected T-join problem. Next, we improve on the approximation guarantee of \(\frac{5}{3}\) by extending their LP-rounding algorithm to get an approximation guarantee of \(\frac{13}{8}=1.625\) for all |T|≥4.
Finally, we focus on the prize-collecting version of the problem, and present a primal-dual algorithm that is “Lagrangian multiplier preserving” and that achieves an approximation guarantee of \(3-\frac{4}{|T|}\) when |T|≥4. Our primal-dual algorithm is a generalization of the known primal-dual 2-approximation for the prize-collecting s,t path TSP. Furthermore, we show that our analysis is tight by presenting instances with |T|≥4 such that the cost of the solution found by the algorithm is exactly \(3-\frac{4}{|T|}\) times the cost of the constructed dual solution.
This is a preview of subscription content, log in to check access.
References
- 1.
An, H.-C., Kleinberg, R., Shmoys, D.B.: Improving Christofides’ algorithm for the s-t path TSP. In: Proceedings of the 44th Annual ACM Symposium on Theory of Computing, pp. 875–886 (2012). CoRR arXiv:1110.4604v2 (2011)
- 2.
Archer, A., Bateni, M., Hajiaghayi, M., Karloff, H.J.: Improved approximation algorithms for prize-collecting Steiner tree and TSP. SIAM J. Comput. 40(2), 309–332 (2011)
- 3.
Balas, E.: The prize-collecting traveling salesman problem. Networks 19(6), 621–636 (1989)
- 4.
Barahona, F., Conforti, M.: A construction for binary matroids. Discrete Math. 66(3), 213–218 (1987)
- 5.
Chaudhuri, K., Godfrey, B., Rao, S., Talwar, K.: Paths, trees, and minimum latency tours. In: Proceedings of the 44nd Annual IEEE Symposium on Foundations of Computer Science, pp. 36–45 (2003)
- 6.
Christofides, N.: Worst-case analysis of a new heuristic for the travelling salesman problem. Technical report. Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, PA (1976)
- 7.
Chudak, F.A., Roughgarden, T., Williamson, D.P.: Approximate k-MSTs and k-Steiner trees via the primal-dual method and Lagrangean relaxation. Math. Program. 100(2), 411–421 (2004)
- 8.
Edmonds, J., Johnson, E.: Matching: a well-solved class of integer linear programs. In: Guy, R., et al. (eds.) Proceedings of the Calgary International Conference on Combinatorial Structures and Their Applications, pp. 82–92. Gordon & Breach, New York (1970)
- 9.
Goemans, M.X.: Combining approximation algorithms for the prize-collecting TSP. CoRR arXiv:0910.0553 (2009)
- 10.
Goemans, M.X., Williamson, D.P.: A general approximation technique for constrained forest problems. SIAM J. Comput. 24(2), 296–317 (1995)
- 11.
Hoogeveen, J.A.: Analysis of Christofides’ heuristic: some paths are more difficult than cycles. Oper. Res. Lett. 10, 291–295 (1991)
- 12.
Jain, K., Vazirani, V.V.: Approximation algorithms for metric facility location and k-Median problems using the primal-dual schema and Lagrangian relaxation. J. ACM 48(2), 274–296 (2001)
- 13.
Lau, L.C., Ravi, R., Singh, M.: Iterative Methods in Combinatorial Optimization. Cambridge University Press, Cambridge (2011)
- 14.
Mömke, T., Svensson, O.: Approximating graphic TSP by matchings. In: Proceedings of the 52nd Annual Symposium on Foundations of Computer Science, pp. 560–569 (2011). CoRR arXiv:1104.3090 (2011)
- 15.
Mucha, M.: 13/9-Approximation for graphic TSP. In: Proceedings of the 29th International Symposium on Theoretical Aspects of Computer Science, pp. 30–41 (2012). Improved analysis for graphic TSP approximation via matchings. CoRR arXiv:1108.1130 (2011)
- 16.
Oveis Gharan, S., Saberi, A., Singh, M.: A randomized rounding approach to the traveling salesman problem. In: Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science, pp. 550–559 (2011)
- 17.
Sebő, A.: Eight-fifth approximation for the path TSP. In: Integer Programming and Combinatorial Optimization. Lecture Notes in Computer Science, vol. 7801, pp. 362–374. Springer, Berlin (2013)
- 18.
Sebő, A., Vygen, J.: Shorter tours by nicer ears: 7/5-approximation for graphic TSP, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs. CoRR arXiv:1201.1870v2 (2012)
- 19.
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, Algorithms and Combinatorics vol. 24. Springer, Berlin (2003)
- 20.
Williamson, D.P., Shmoys, D.B.: The Design of Approximation Algorithms. Cambridge University Press, Cambridge (2011)
Acknowledgements
We thank a number of colleagues for useful discussions; in particular, we thank Jochen Könemann and Chaitanya Swamy. We thank two anonymous reviewers for their comments; these comments resulted in several improvements.
Author information
Additional information
An extended abstract of this work has been published in APPROX–RANDOM 2012, LNCS 7408, pp. 110–121.
J. Cheriyan supported by NSERC grant No. OGP0138432.
Rights and permissions
About this article
Cite this article
Cheriyan, J., Friggstad, Z. & Gao, Z. Approximating Minimum-Cost Connected T-Joins. Algorithmica 72, 126–147 (2015). https://doi.org/10.1007/s00453-013-9850-8
Received:
Accepted:
Published:
Issue Date:
Keywords
- Approximation algorithms
- LP rounding
- Primal-dual method
- Prize-collecting problems
- T-Joins
- Traveling Salesman Problem
- s,t-Path TSP