The Steiner Traveling Salesman Problem (STSP) is a variant of the classical Traveling Salesman Problem. The STSP involves incorporating steiner nodes, which are extra nodes not originally part of the required visit set but that can be added to the route to enhance the overall solution and minimize the total travel cost. Given the NP-hard nature of the STSP, we propose a quantum approach to address it. Specifically, we employ quantum annealing using D-Wave's hardware to explore its potential for solving this problem. To enhance computational feasibility, we develop a preprocessing method that effectively reduces the network size. Our experimental results demonstrate that this reduction technique significantly decreases the problem complexity, making the Quadratic Unconstrained Binary Optimization formulation, the standard input for quantum annealers, better suited for existing quantum hardware. Furthermore, the results highlight the potential of quantum annealing as a promising and innovative approach for solving the STSP.

The Rectilinear Steiner Minimum Tree (RSMT) problem is a fundamental problem in VLSI placement and routing and is known to be NP-hard. Traditional RSMT algorithms spend a significant amount of time on finding Steiner points to reduce the total wire length or use heuristics to approximate producing sub-optimal results. We show that Graph Neural Networks (GNNs) can be used to predict optimal Steiner points in RSMTs with high accuracy and can be parallelized on GPUs. In this paper, we propose GAT-Steiner, a graph attention network model that correctly predicts 99.846% of the nets in the ISPD19 benchmark with an average increase in wire length of only 0.480% on suboptimal wire length nets. On randomly generated benchmarks, GAT-Steiner correctly predicts 99.942% with an average increase in wire length of only 0.420% on suboptimal wire length nets.

Given data, finding a faithful low-dimensional hyperbolic embedding of the data is a key method by which we can extract hierarchical information or learn representative geometric features of the data. In this paper, we explore a new method for learning hyperbolic representations by taking a metric-first approach. Rather than determining the low-dimensional hyperbolic embedding directly, we learn a tree structure on the data. This tree structure can then be used directly to extract hierarchical information, embedded into a hyperbolic manifold using Sarkar's construction \cite{sarkar}, or used as a tree approximation of the original metric. To this end, we present a novel fast algorithm \textsc{TreeRep} such that, given a $\delta$-hyperbolic metric (for any $\delta \geq 0$), the algorithm learns a tree structure that approximates the original metric. In the case when $\delta = 0$, we show analytically that \textsc{TreeRep} exactly recovers the original tree structure. We show empirically that \textsc{TreeRep} is not only many orders of magnitude faster than previously known algorithms, but also produces metrics with lower average distortion and higher mean average precision than most previous algorithms for learning hyperbolic embeddings, extracting hierarchical information, and approximating metrics via tree metrics.

The recent years and events lead to a massive development of content-oriented cloud services. The most popular and voluminous content o¤ered in today's networks are videos that must be e¢ ciently delivered to end customers. The objective of the service provider (root) is to optimize the delivery of content to its costumers (terminals). In this optimization problem the cost is usually assumed to be known (left graph). Yet, in reality it is often unknown because it depends on many stochastic factors, such as the tra¢ c on the network, the level of demand, and so on (right graph). Figure 1: Graphical representation of networks where information travels from a root to a set of terminals over channels with known or unknown cost.

The Steiner Tree Problem (STP) is an important problem in combinatorial optimization which has numerous applications, ranging from the design of (very large) integrated circuits to computer networking, evolution theory in biology and more [8]. There are plenty variants of the STP which can be found in [7]. The common part between different variants is the requirement to connect a set of objects with the shortest interconnect possible. In this paper, we investigate the general STP in graphs. As the STP is N P-hard [10], most of the work in the literature focuses on non-exact approaches.