The expansion of Fiber-To-The-Home (FTTH) networks creates high costs due to expensive excavation procedures. Optimizing the planning process and minimizing the cost of the earth excavation work therefore lead to large savings. Mathematically, the FTTH network problem can be described as a minimum Steiner Tree problem. Even though the Steiner Tree problem has already been investigated intensively in the last decades, it might be further optimized with the help of new computing paradigms and emerging approaches. This work studies upcoming technologies, such as Quantum Annealing, Simulated Annealing and nature-inspired methods like Evolutionary Algorithms or slime-mold-based optimization. Additionally, we investigate partitioning and simplifying methods. Evaluated on several real-life problem instances, we could outperform a traditional, widely-used baseline (NetworkX Approximate Solver) on most of the domains. Prior partitioning of the initial graph and the presented slime-mold-based approach were especially valuable for a cost-efficient approximation. Quantum Annealing seems promising, but was limited by the number of available qubits.