Advertisement

# Deadline TSP

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10787)

## 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 ) 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. ). We also present improved bi-criteria $$(1+\varepsilon ,\frac{1}{1+\varepsilon })$$-approximation algorithms for the Deadline TSP on weighted trees.

## References

1. 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. 2.
Bockenhauer, H., Hromkovic, J., Kneis, J., Kupke, J.: The parameterized approximability of TSP with deadlines. Theory Comput. Syst. 41(3), 431–444 (2007)
3. 3.
Chekuri, C., Korula, N.: Approximation algorithms for orienteering with time windows. CoRR, abs/0711.4825 (2007)Google Scholar
4. 4.
Chekuri, C., Korula, N., Pál, M.: Improved algorithms for orienteering and related problems. ACM Trans. Algorithms 8(3), 23:1–23:27 (2012)
5. 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. 6.
Frederickson, G.N., Wittman, B.: Approximation algorithms for the traveling repairman and speeding deliveryman problems. Algorithmica 62(3), 1198–1221 (2012)
7. 7.
Hochbaum, D.S. (ed.): Approximation Algorithms for NP-Hard Problems. PWS Publishing Co., Boston (1997)
8. 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)

## Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

## Authors and Affiliations

1. 1.Faculty of Industrial Engineering and ManagementThe TechnionHaifaIsrael