Algorithmica

, Volume 78, Issue 1, pp 298–318

# The Maximum Labeled Path Problem

• Basile Couëtoux
• Elie Nakache
• Yann Vaxès
Article

## Abstract

In this paper, we study the approximability of the Maximum Labeled Path problem: given a vertex-labeled directed acyclic graph D, find a path in D that collects a maximum number of distinct labels. For any $$\epsilon >0$$, we provide a polynomial time approximation algorithm that computes a solution of value at least $$OPT^{1-\epsilon }$$ and a self-reduction showing that any constant ratio approximation algorithm for this problem can be converted into a PTAS. This last result, combined with the APX-hardness of the problem, shows that the problem cannot be approximated within any constant ratio unless $$P=NP$$.

## Keywords

Graph algorithm Approximation algorithm Hardness of approximation Labels

## Notes

### Acknowledgments

We are grateful to Jérôme Monnot for suggesting the use of a self-reduction to prove the hardness result of Sect. 3.

## References

1. 1.
Batten, L.M.: Combinatorics of Finite Geometries. Cambridge University Press, New York (1997)
2. 2.
Broersma, H., Li, X.: Spanning trees with many or few colors in edge-colored graphs. Discuss. Math. Graph Theory 17, 259–269 (1997)
3. 3.
Broersma, H., Li, X., Woeginger, G.J., Zhang, S.: Paths and cycles in colored graphs. Australas. J. Comb. 31, 299–311 (2005)
4. 4.
Brüggemann, T., Monnot, J., Woeginger, G.J.: Local search for the minimum label spanning tree problem with bounded color classes. Oper. Res. Lett. 31, 195–201 (2003)
5. 5.
Chang, R.-S., Leu, S.-J.: The minimum labeling spanning trees. Inf. Process. Lett. 31, 195–201 (2003)Google Scholar
6. 6.
Couëtoux, B., Gourvès, L., Monnot, J., Telelis, O.: Labeled traveling salesman problems: complexity and approximation. Discrete Optim. 7, 74–85 (2010)
7. 7.
Hassin, R., Monnot, J., Segev, D.: Approximation algorithms and hardness results for labeled connectivity problems. J. Comb. Optim. 14, 437–453 (2007)
8. 8.
Hassin, R., Monnot, J., Segev, D.: The complexity of bottleneck labeled graph problems. In: International Workshop on Graph-Theoretic Concepts in Computer Science (WG). Lecture Notes in Computer Science, vol. 4769. Springer, Berlin (2007)Google Scholar
9. 9.
Håstad, J.: Some optimal inapproximability results. J. ACM 48, 798–859 (2001)
10. 10.
Krumke, S.O., Wirth, H.-C.: Approximation algorithms and hardness results for labeled connectivity problems. Inf. Process. Lett. 66, 81–85 (1998)
11. 11.
Monnot, J.: The labeled perfect matching in bipartite graphs. Inf. Process. Lett. 96, 81–88 (2005)