Star Routing: Between Vehicle Routing and Vertex Cover
We consider an optimization problem posed by an actual newspaper company, which consists of computing a minimum length route for a delivery truck, such that the driver only stops at street crossings, each time delivering copies to all customers adjacent to the crossing. This can be modeled as an abstract problem that takes an unweighted simple graph \(G = (V, E)\) and a subset of edges X and asks for a shortest cycle, not necessarily simple, such that every edge of X has an endpoint in the cycle.
We show that the decision version of the problem is strongly NP-complete, even if G is a grid graph. Regarding approximate solutions, we show that the general case of the problem is APX-hard, and thus no PTAS is possible unless \( P = NP \). Despite the hardness of approximation, we show that given any \(\alpha \)-approximation algorithm for metric TSP, we can build a \(3\alpha \)-approximation algorithm for our optimization problem, yielding a concrete 9 / 2-approximation algorithm.
The grid case is of particular importance, because it models a city map or some part of it. A usual scenario is having some neighborhood full of customers, which translates as an instance of the abstract problem where almost every edge of G is in X. We model this property as \(|E - X| = o(|E|)\), and for these instances we give a \((3/2 + \varepsilon )\)-approximation algorithm, for any \(\varepsilon > 0\), provided that the grid is sufficiently big.
KeywordsVehicle routing Vertex cover Approximation algorithms Computational complexity
Thanks to Martín Farach-Colton for useful discussions and suggestions about the presentation.
- 2.de Berg, M., Gudmundsson, J., Katz, M.J., Levcopoulos, C., Overmars, M.H., van der Stappen, A.F.: TSP with neighborhoods of varying size. J. Algorithms 57(1), 22–36 (2005). https://doi.org/10.1016/j.jalgor.2005.01.010. http://www.sciencedirect.com/science/article/pii/S0196677405000246MathSciNetCrossRefzbMATHGoogle Scholar
- 3.Christofides, N.: Worst-case analysis of a new heuristic for the travelling salesman problem. Technical report 388, Graduate School of Industrial Administration, Carnegie Mellon University (1976)Google Scholar
- 7.Demaine, E.D., Rudoy, M.: A simple proof that the \((n^2-1)\)-puzzle is hard. Computing Research Repository abs/1707.03146 (2017)Google Scholar
- 9.Dumitrescu, A., Mitchell, J.S.B.: Approximation algorithms for tsp with neighborhoods in the plane. J. Algorithms 48(1), 135–159 (2003). https://doi.org/10.1016/S0196-6774(03)00047-6. http://www.sciencedirect.com/science/article/pii/S0196677403000476. Twelfth Annual ACM-SIAM Symposium on Discrete AlgorithmsMathSciNetCrossRefzbMATHGoogle Scholar
- 10.Garey, M.R., Graham, R.L., Johnson, D.S.: Some NP-complete geometric problems. In: Proceedings of the Eighth Annual ACM Symposium on Theory of Computing, pp. 10–22. STOC 1976. ACM, New York, NY, USA (1976)Google Scholar
- 12.Grigni, M., Koutsoupias, E., Papadimitriou, C.: An approximation scheme for planar graph TSP. In: Proceedings of the 36th Annual Symposium on Foundations of Computer Science. p. 640. FOCS 1995. IEEE Computer Society, Washington, DC, USA (1995)Google Scholar