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The Rectilinear Steiner Tree Problem is $NP$-Complete
Abstract
An optimum rectilinear Steiner tree for a set A of points in the plane is a tree which interconnects A using horizontal and vertical lines of shortest possible total length. Such trees correspond to single net wiring patterns on printed backplanes which minimize total wire length. We show that the problem of determining this minimum length, given A, is $NP$-complete. Thus the problem of finding optimum rectilinear Steiner trees is probably computationally hopeless, and the emphasis of the literature for this problem on heuristics and special case algorithms is well justified. A number of intermediary lemmas concerning the $NP$-completeness of certain graph-theoretic problems are proved and may be of independent interest.
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Cited by
GRANT, T., FROST, D. R., CALDWELL, J. P., GAGLIARDO, R., HADDAD, C. F. B., KOK, P. J. R., MEANS, D. B., NOONAN, B. P., SCHARGEL, W. E., & WHEELER, W. C. (2006) · Bulletin of the American Museum of Natural History · doi:10.1206/0003-0090(2006)299[1:psodfa]2.0.co;2 · Cited by 578
KagNet: Knowledge-Aware Graph Networks for Commonsense Reasoning
Lin, B. Y., Chen, X., Chen, J., & Ren, X. (2019) · Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP) · doi:10.18653/v1/d19-1282 · Cited by 452
Quantum optimization of maximum independent set using Rydberg atom arrays
Ebadi, S., Keesling, A., Cain, M., Wang, T. T., Levine, H., Bluvstein, D., Semeghini, G., Omran, A., Liu, J.-G., Samajdar, R., et al. (2022) · Science · doi:10.1126/science.abo6587 · Cited by 430
An improved LP-based approximation for steiner tree
Byrka, J., Grandoni, F., Rothvoß, T., & Sanità, L. (2010) · Proceedings of the forty-second ACM symposium on Theory of computing · doi:10.1145/1806689.1806769 · Cited by 296
Power outage prediction for natural hazards using synthetic power distribution systems
Zhai, C., Chen, T. Y.-J., White, A. G., & Guikema, S. D. (2021) · Reliability Engineering & System Safety · doi:10.1016/j.ress.2020.107348 · Cited by 64
Metadata
- Authors
- Garey, M. R., & Johnson, D. S.
- Venue
- SIAM Journal on Applied Mathematics
- Published
- 1977-06
- Volume
- 32
- Issue
- 4
- Pages
- 826-834
- ISSN
- 0036-1399 (ISSN-L)
- 1095-712X
- OpenAlex
- W2020112047
- Cited by
- 1,124
- References
- 0
- FWCI
- 8.24
- Type
- journal article
- Language
- en
- Open access
- closed
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- Garey, M. R., & Johnson, D. S. (1977). The Rectilinear Steiner Tree Problem is $NP$-Complete. SIAM Journal on Applied Mathematics. https://doi.org/10.1137/0132071
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