Reducing dense virtual networks for fast embedding

Virtual network embedding has been intensively studied for a decade. The time complexity of most conventional methods has been reduced to the cube of the number of links. Since customers are likely to request a dense virtual network that connects every node pair directly (|E| = O(|V|2)) based on a traffic matrix, the time complexity is actually O(|E|3 = |V|6). If we were allowed to reduce this dense network into a sparse one before embedding, the time complexity could be decreased to O(|V|3); the time gap can be a million times for |V| = 100. The network reduction, however, combines several virtual links into a broader link, which makes the embedding cost (solution quality) much worse. This paper analytically and empirically investigates the trade-off between the embedding time and cost for the virtual network reduction. We define two simple reduction algorithms and analyze them with several interesting theorems. The analysis indicates that the embedding cost increases only linearly with exponential decay of embedding time. Thorough numerical evaluation justifies the desirability of the trade-off.