Virtual Network Embedding via Decomposable LP Formulations: Orientations of Small Extraction Width and Beyond
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The Virtual Network Embedding Problem (VNEP) considers the efficient allocation of resources distributed in a substrate network to a set of request networks. Many existing works discuss either heuristics or exact algorithms, resulting in a choice between quick runtimes and quality guarantees for the solutions. Recently, the first fixed-parameter tractable (FPT) approximation algorithm for the VNEP with arbitrary request and substrate topologies has been published by Rost and Schmid. This algorithm is based on a LP formulation and is FPT in the newly introduced graph parameter extraction width (EW). It therefore combines positive traits of heuristics and exact approaches: The runtime is polynomial for instances with bounded EW, and the algorithm returns approximate solutions with high probability. We propose two extensions of this algorithm to optimize its runtime. Firstly, we develop a new LP formulation related to tree decompositions. We show that this LP formulation is always smaller, and that the resulting algorithm is FPT in the new parameter extraction label width (ELW). We improve on two important results by Rost and Schmid: For centrally rooted half-wheel topologies, the EW scales linearly with request size, whereas the ELW is constant. Further, adding any number of paths parallel to an existing edge increases the EW by at most the maximum degree of the request. By contrast, the ELW only increases by at most one. Lastly, we show that finding the minimal ELW is NP-hard. Secondly, we consider the approach of partitioning the request into subgraphs which are processed independently, yielding even smaller LP formulations. While this algorithm may lead to higher ELW within the subgraphs, we show that this increase is always smaller than the size of the inter-subgraph boundary. In particular, the algorithm has zero additional cost when subgraphs are separated by a single node.
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