# Deadline TSP

## Abstract

We study the Deadline TSP problem. The input consists of a complete undirected graph \(G=(V,E)\), a metric \(c:E \rightarrow \mathbf {Z}_+\), a reward function \(w:V\rightarrow \mathbf {Z}_+\), a non-negative deadline function \(d:V\rightarrow \mathbf {Z}_+\), and a starting node \(s \in V\). A feasible solution is a path starting at *s*. Given such a path and a node \(v\in V\), we say that the path visits *v* by its deadline if the length of the prefix of the path starting at *s* until the first time it traverses *v* is at most *d*(*v*) (in particular, it means that the path traverses *v*). If a path visits *v* by its deadline, it gains the reward *w*(*v*). The objective is to find a path *P* starting at *s* that maximizes the total reward. In our work we present a bi-criteria (\(1+\varepsilon ,\frac{\alpha }{1+\varepsilon }\))-approximation algorithm for every \(\varepsilon >0\) for the Deadline TSP, where \(\alpha \) is the approximation ratio for Deadline TSP with a constant number of deadlines (currently \(\alpha =\frac{1}{3}\) by [5]) and thus significantly improving the previously best known bi-criteria approximation for that problem (a bi-criteria \((1+\varepsilon ,\frac{1}{O(\log (1/\varepsilon ))})\)-approximation algorithm for every \(\varepsilon >0\) by Bansal et al. [1]). We also present improved bi-criteria \((1+\varepsilon ,\frac{1}{1+\varepsilon })\)-approximation algorithms for the Deadline TSP on weighted trees.

## References

- 1.Bansal, N., Blum, A., Chawla, S., Meyerson, A.: Approximation algorithms for deadline-TSP and vehicle routing with time-windows. In: Proceedings of STOC 2004, pp. 166–174 (2004)Google Scholar
- 2.Bockenhauer, H., Hromkovic, J., Kneis, J., Kupke, J.: The parameterized approximability of TSP with deadlines. Theory Comput. Syst.
**41**(3), 431–444 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Chekuri, C., Korula, N.: Approximation algorithms for orienteering with time windows. CoRR, abs/0711.4825 (2007)Google Scholar
- 4.Chekuri, C., Korula, N., Pál, M.: Improved algorithms for orienteering and related problems. ACM Trans. Algorithms
**8**(3), 23:1–23:27 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Chekuri, C., Kumar, A.: Maximum coverage problem with group budget constraints and applications. In: Proceedings of APPROX 2004, pp. 72–83 (2004)Google Scholar
- 6.Frederickson, G.N., Wittman, B.: Approximation algorithms for the traveling repairman and speeding deliveryman problems. Algorithmica
**62**(3), 1198–1221 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Hochbaum, D.S. (ed.): Approximation Algorithms for NP-Hard Problems. PWS Publishing Co., Boston (1997)zbMATHGoogle Scholar
- 8.Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM
**32**(1), 130–136 (1985)MathSciNetCrossRefzbMATHGoogle Scholar