Minimum routing cost spanning tree

In computer science, the minimum routing cost spanning tree of a weighted graph is a spanning tree minimizing the sum of pairwise distances between vertices in the tree. It is also called the optimum distance spanning tree, shortest total path length spanning tree, minimum total distance spanning tree, or minimum average distance spanning tree. In an unweighted graph, this is the spanning tree of minimum Wiener index.[1] Hu (1974) writes that the problem of constructing these trees was proposed by Francesco Maffioli.[2]

It is NP-hard to construct it, even for unweighted graphs.[3][4] However, it has a polynomial-time approximation scheme. The approximation works by choosing a number  k {\displaystyle k} that depends on the approximation ratio but not on the number of vertices of the input graph, and by searching among all trees with  k {\displaystyle k} internal nodes.[5]

The minimum routing cost spanning tree of an unweighted interval graph can be constructed in linear time.[6] A polynomial time algorithm is also known for distance-hereditary graphs, weighted so that the weighted distances are hereditary.[7]

## References[edit]

1. ^ Dobrynin, Andrey A.; Entringer, Roger; Gutman, Ivan (2001). "Wiener index of trees: theory and applications". Acta Applicandae Mathematicae. 66 (3): 211–249. doi:10.1023/A:1010767517079. MR 1843259. S2CID 116000819.
2. ^ Hu, T. C. (1974). "Optimum communication spanning trees". SIAM Journal on Computing. 3 (3): 188–195. doi:10.1137/0203015. MR 0427116.
3. ^ Johnson, D. S.; Lenstra, J. K.; Rinnooy Kan, A. H. G. (1978). "The complexity of the network design problem" (PDF). Networks. 8 (4): 279–285. doi:10.1002/net.3230080402. MR 0514463.
4. ^ Michael R. Garey and David S. Johnson (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman. ISBN 0-7167-1045-5. A2.1: ND3, pg.206.
5. ^ Wu, Bang Ye; Lancia, Giuseppe; Bafna, Vineet; Chao, Kun-Mao; Ravi, R.; Tang, Chuan Yi (2000). "A polynomial-time approximation scheme for minimum routing cost spanning trees" (PDF). SIAM Journal on Computing. 29 (3): 761–778. doi:10.1137/S009753979732253X. MR 1740574. S2CID 1639329.
6. ^ Dahlhaus, Elias; Dankelmann, Peter; Ravi, R. (2004). "A linear-time algorithm to compute a MAD tree of an interval graph". Information Processing Letters. 89 (5): 255–259. doi:10.1016/j.ipl.2003.11.009. MR 2032568.
7. ^ Dahlhaus, E.; Dankelmann, P.; Goddard, W.; Swart, H. C. (2003). "MAD trees and distance-hereditary graphs". Discrete Applied Mathematics. 131 (1): 151–167. CiteSeerX 10.1.1.86.6022. doi:10.1016/S0166-218X(02)00422-5. MR 2016490.

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