Acta Mathematicae Applicatae Sinica, English Series
Vol. 33, No. 3 (2017) 561–566
http://www.ApplMath.com.cn & www.SpringerLink.com
Acta MathemaƟcae Applicatae Sinica,
The Editorial Office of AMAS &
Springer-Verlag Berlin Heidelberg 2017
New Discoveries of Domination Between Traﬃc Matrices
, Tian-de GUO
School of Mathematics Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
Key Laboratory of Big Data Mining and Knowledge Management, Chinese Academy of Sciences, Beijing
Abstract In this paper the deﬁnition of domination is generalized to the case that the elements of the traﬃc
matrices may have negative values. It is proved that D
) for any λ
.LetU(D) be the set of all the traﬃc matrices that are dominated by the traﬃc matrix D.Itis
shown that U(D
) are isomorphic. Besides, similar results are obtained on multi-commodity ﬂow
problems. Furthermore, the results are the generalized to integral ﬂows.
Keywords robust network design; multi-commodity ﬂows; domination
2000 MR Subject Classiﬁcation 90B10
Network design problem is a classical problem. Given a graph with capacity installation costs
for the edges and a set of pairwise traﬃc demands— a traﬃc matrix, the problem consists of
choosing minimum cost capacities such that all the demands can be routed simultaneously on
However, it is too restrictive under the assumption that there is only one traﬃc matrix
to be considered, and that this matrix is known in advance or can be reliably estimated.
Unfortunately, in several applications, demands change over time and can be diﬃcult to predict.
Therefore, for the sake of obtaining a more ﬂexible and robust network, it is necessary to take
the demand uncertainty into consideration in the design process. A possible way is to consider
a set of traﬃc matrices to be served non-simultaneously, instead of a single traﬃc matrix. This
version of the problem is known as the Robust Network Design problem(RND), which has
attracted a great deal of attention
In  the author introduces the concept of domination. It is said D
capacity reservation supporting D
also supports D
. The author points out that D
if and only if D
, regarded as a capacity reservation, supports D
We generalize the deﬁnition of domination to the case that the elements of the traﬃc
matrices may take negative values. One of our main results is a good property of domination: let
be three traﬃc matrices, if D
for any λ ≥ 0 (Theorem 3.1). To the best of our knowledge this simple but surprising result
has not been observed before. The direct application of this property leads to another equally
surprising result. Let U(D) be the set of all traﬃc matrices that are dominated by the traﬃc
) are isomorphic. Then we generalize these
results to multi-commodity ﬂow problems.
Manuscript received January 25, 2015. Revised December 28, 2015.
Supported by National Natural Science Foundation of China under Grant No.(1157101511331012), the Open
Project of Key Laboratory of Big Data Mining and Knowledge Management and Knowledge Innovation Program
of the Chinese Academy of Sciences under Grant No.(KGCX2-RW-329).