Traffic engineering (TE) is a fundamental task in networking. Conventionally, traffic can take any path connecting the source and destination. Emerging technologies such as segment routing, however, use logical paths going through a predetermined set of middlepoints. Inspired by this, in this work we introduce the problem of node-constrained TE, where traffic must go through a set of middlepoints, and study its theoretical fundamentals. We show that the general node-constrained TE that constrains the traffic to take paths going through one or more middlepoints is NP-hard for directed graphs but strongly polynomial for undirected graphs, unveiling a profound dichotomy between the two cases. We additionally investigate the popular variant of node-constrained TE that uses shortest paths between middlepoints, and show that the problem can now be solved in weakly polynomial time for a fixed number of middlepoints. Yet if we constrain the end-to-end paths to be acyclic, the problem can become NP-hard. This explains why existing work focuses on the computationally tractable variant. An important application of our work concerns the computational complexity of flow centrality, first proposed in 1991 by Freeman et al. [21]: we show that it is NP-hard for directed but strongly polynomial for undirected graphs. Finally, we investigate the middlepoint selection problem in general node-constrained TE. We introduce group flow centrality as a solution concept for multi-commodity networks, study its complexity, and show that it is monotone but not submodular for both directed and undirected graphs. Our work provides a thorough theoretical treatment of nodeconstrained TE and its applications.
We assume a data set that is vertically decomposed among several servers, and a client that wishes to compute the skyline by obtaining the minimum number of points. Existing solutions for this problem are restricted to the case where each server maintains exactly one dimension. This paper proposes a general solution for vertical decompositions of arbitrary dimensionality. We first investigate some interesting problem characteristics regarding the pruning power of points. Then, we introduce vertical partition skyline (VPS), an algorithmic framework that includes two steps. Phase 1 searches for an anchor point P anc that dominates, and hence eliminates, a large number of records. Starting with P anc , Phase 2 constructs incrementally a pruning area using an interesting unionintersection property of dominance regions. Servers do not transmit points that fall within the pruning area in their local subspace. Our experiments confirm the effectiveness of the proposed methods under various settings.
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