In this paper, we take an in-depth look at the complexity of a hitherto unexplored multiobjective minimum weight minimum stretch spanner problem; or in short multiobjective spanner (MSp) problem. The MSp is a multiobjective generalization of the well-studied minimum t-spanner problem. This multiobjective approach allows to find solutions that offer a viable compromise between cost and utility—a property that is usually neglected in singleobjective optimization. Thus, the MSp can be a powerful modeling tool when it comes to, e.g., the planning of transportation or communication networks. This holds especially in disaster management, where both responsiveness and practicality are crucial. We show that for degree-3 bounded outerplanar instances the MSp is intractable and computing the non-dominated set is BUCO-hard. Additionally, we prove that if \({\textbf{P}} \ne \textbf{NP}\), the set of extreme points cannot be computed in output-polynomial time, for instances with unit costs and arbitrary graphs. Furthermore, we consider the directed versions of the cases above.

在本文中，我们深入研究了迄今为止尚未探索的多目标最小权重最小拉伸扳手问题的复杂性；或者简而言之，多目标扳手（MSp）问题。 MSp 是经过充分研究的最小 t 扳手问题的多目标推广。这种多目标方法可以找到在成本和效用之间提供可行折衷的解决方案，而这一特性在单目标优化中通常被忽视。因此，在交通或通信网络规划等方面，MSp 可以成为强大的建模工具。这尤其适用于灾害管理，响应性和实用性都至关重要。我们表明，对于 3 度有界外平面实例，MSp 很棘手，并且计算非支配集是BUCO困难的。此外，我们证明，如果\({\textbf{P}} \ne \textbf{NP}\)，对于具有单位成本和任意图的实例，无法在输出多项式时间内计算极值点集。此外，我们考虑上述案例的定向版本。