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We introduce the set of quantum channels with constant Frobenius norm, the set of diagonal channels and the notion of equivalence of oneparameter families of channels. First, we show that all diagonal 2-dimensional channels with constant Frobenius norm are equivalent. Next, we generalize four one-parameter families of 2-dimensional diagonal channels with constant Frobenius norm to an arbitrary dimension n. Finally, we prove that the generalizations are not equivalent in any dimension n ≥ 3.

We investigate network coding (NC) in access point-to-multipoint (PMP) broadcast networks. Characterized by a shared unicast upstream channel and a time-shared broadcast downstream channel, PMP networks are widely deployed in optical and wireless access networks. We develop a queuing-theoretic model of NC at the medium access control (MAC) sublayer and analyze the impact of NC on packet delay. Our analysis is validated through discrete-event simulation and demonstrates significant delay advantages for NC under high loads and localized traffic.

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The k-Center problem is one of the most popular clustering problems. After decades of work, the complexity of most of its variants on general metrics is now well understood. Surprisingly, this is not the case for a natural setting that often arises in practice, namely the Euclidean setting, in which the input points are points in R d , and the distance between them is the standard ℓ 2 Euclidean distance. In this work, we study two Euclidean k-Center variants, the Matroid Center problem on the real line and the Robust Euclidean k-Supplier problem, and provide algorithms that improve upon the best approximation guarantees known for these problems.The Matroid Center problem on the real line is one of the rare instances of a 1-dimensional k-Center variant that is NP-hard, as shown by Chen, Li, Liang, and Wang (2016); most k-Center problems become easy when restricted to the real line. In fact, Chen et al. showed that the problem is (2 − ε)-hard to approximate. On the algorithmic side, only the 3-approximation algorithm for Matroid Center on general metrics by Chen et al. is known for tackling the problem. In this work, building on the classic threshold technique of Hochbaum and Shmoys (1986) and by exploiting the very special structure of real-line metrics, we improve upon the 3-approximation factor and provide a simple 2.5-approximation algorithm.We then turn to the Robust k-Supplier problem (also known as k-Supplier with outliers), which is one of the most popular k-Center variants that have been studied in the literature. It is known that the problem admits a 3-approximation on general metrics, which is tight even when there are no outliers, assuming P = NP. We focus on the Euclidean setting, for which the 3 − ε hardness does not hold anymore. For the special case where there are no outliers, Nagarajan, Schieber and Shachnai (2020) gave a very elegant (1 + √ 3)-approximation algorithm for the Euclidean k-Supplier problem, thus overcoming the 3 − ε barrier. However, their algorithm does not generalize to the robust setting. In this work, building on the ideas of Nagarajan et al. and the general round-or-cut framework of Chakrabarty and Negahbani (2019) that gives tight 3approximation algorithms for many Robust k-Center variants on general metrics, we present a (1 + √ 3)-approximation algorithm for the Robust Euclidean k-Supplier problem, thus improving upon the aforementioned 3-approximation algorithm for Robust k-Supplier on general metrics and matching the best approximation factor known for the non-robust setting. * The results of this work first appeared in the MSc thesis of the third author, where the first two authors served as co-advisors. An independent recent work of Lee, Nagarajan and Wang [12] obtained a (1 + √ 3)-approximation algorithm for Robust Euclidean k-Supplier by using similar, but not identical, ideas.