Abstract
Due to their ubiquity and extensive applications, graph routing problems have been widely studied in the fields of operations research, computer science and engineering. An important example of a routing problem is the traveling salesman problem. In this paper, we consider two variants of the general cluster routing problem. In these variants, we are given a complete undirected graph \(G=(V,E)\) with a metric cost function c and a partition \(V=C_{1}\cup C_{2}\cdots \cup C_{k}\) of the vertex set. For a given subset \(V^{'}\) of V and subset \(E^{'}\) of E, the task is to find a walk T with minimum cost such that it visits each vertex in \(V^{\prime }\) exactly once and covers each edge in \(E^{\prime }\) at least once. Besides, for every \(i\in [k]\), all the vertices in \(T\cap C_i\) are required to be visited consecutively in T. We design two combinatorial approximation algorithms with ratios 21/11 and 2.75 for the two variants, respectively; both ratios match the approximation ratios for the corresponding variants of the cluster traveling salesman problem, a special case of general cluster routing problem.
Similar content being viewed by others
References
An H-C, Kleinberg R, Shmoys D-B (2015) Improving Christofides’ algorithm for the \(s\)-\(t\) path TSP. J ACM 62(5):34
Anily S, Bramel J-D, Hertz A (1999) A 5/3-approximation algorithm for the clustered traveling salesman tour and path problems. Oper Res Lett 24(1–2):29–35
Arkin E, Hassin R, Klein L (1997) Restricted delivery problems on a network. Networks 29:205–216
Bienstock D, Goemans M-X, Simchi D, Williamson D-P (1991) A note on the prize-collecting traveling salesman problem. Math Program 59:413–420
Christofides N (1976) Worst-case analysis of a new heuristic for the traveling salesman problem. Carnegie-Mellon Univ Pittsburgh Pa Management Sciences Research Group
Frederickson G-N (1979) Approximation algorithms for some postman problems. J ACM 26:538–554
Fumei L, Alantha N (2008) Traveling salesman path problems. Math Progrom 13:39–59
Golden B-L (2010) A statistical approach to the TSP. Networks 7(3):209–225
Gutin G, Punnen A (2002) The traveling salesman problem and its variations. Kluwer, Dordrecht
Gutin G, Yeo A, Zverovich A (2002) Traveling salesman should not be greedy: domination analysis of greedy-type heuristics for the TSP. Discrete Appl Math 117(1–3):81–86
Guttmann-Beck N, Hassin R, Khuller S, Raghavachari B (2000) Approximation algorithms with bounded performance guarantees for the clustered traveling salesman problem. Algorithmica 28:422–437
Hoogeveen J-A (1991) Analysis of Christofides’ heuristic: some paths are more difficult than cycles. Oper Res Lett 10:291–295
Jansen K (1992) An approximation algorithm for the general routing problem. Inf Process Lett 41:333–339
Karp R-M (1972) Reducibility among combinatorial problems. Complex Comput Comput 2:85–103
Lin S, Kernighan B-W (1973) An effective heuristic algorithm for the traveling salesman problem. Oper Res 21(2):498–516
Lovász L (1976) On some connectivity properties of Eulerian multigraphs. Atca Math Acad Sci Hung 28:129–138
Mömke T, Svensson O (2016) Removing and adding edges for the traveling salesman problem. J ACM 63(1):2
Mucha M (2014) 13/9-approximation for graphic TSP. Theory Comput Syst 55:640–657
Punnen A, Kabadi M-S (2003) TSP heuristics: domination analysis and complexity. Algorithmica 35(2):111–127
Sahni S, Gonzales T (1976) \(P\)-complete approximation problems. J ACM 23(3):555–565
Sebö A (1986) Finding thet-join structure of graphs. Math Program 36:123C134
Sebö A (2013) Eight fifth approximation for TSP paths. In: Goemans M, Correa J (eds) Integer Programming and Combinatorial Optimization 2013, LNCS, vol 7801. Springer. Berlin, Heidelberg, pp 362–374
Sebö A, Van Zuylen A (2019) The salesman’s improved paths through forests. J ACM 66(4):28
Sebö A, Vygen J (2014) 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. Combinatorica 34(5):597–629
Traub V, Vygen J (2018) Beating the integrality ratio for \(s\)-\(t\)-tours in graphs. In: Proceedings of 59th annual symposium on foundations of computer science, pp 766–777
Traub V, Vygen J (2019) Approaching \(\frac{3}{2}\) for the \(s\)-\(t\) path TSP. J ACM 66(2):14
Zenklusen R-A (2019) \(1.5\)-Approximation for path TSP. In: Proceedings of the 30th annual ACM-SIAM symposium on discrete algorithms, pp 1539–1549
Zhang X, Du D, Gutin G, Ming Q, Sun J (2020) Approximation algorithms for general cluster routing problem. In: Kim D., Uma R., Cai Z., Lee D. (eds) Computing and combinatorics. COCOON 2020. Lecture notes in computer science, vol 12273. Springer, Cham. https://doi.org/10.1007/978-3-030-58150-3_38
Acknowledgements
This research is supported or partially supported by the National Natural Science Foundation of China (Grant Nos. 11871280, 11771386, 11728104 and 11871081), the Natural Sciences and Engineering Research Council of Canada (NSERC) Grant 06446 and Qinglan Project.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
A preliminary version of this paper was published in the proceedings of 26th International Computing and Combinatorics Conference (COCOON 2020) (Zhang et al. 2020)
Rights and permissions
About this article
Cite this article
Zhang, X., Du, D., Gutin, G. et al. Approximation algorithms with constant ratio for general cluster routing problems. J Comb Optim 44, 2499–2514 (2022). https://doi.org/10.1007/s10878-021-00772-8
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-021-00772-8