We study solution discovery, where the goal is to obtain a feasible solution to a problem from an initial configuration by a bounded sequence of local moves. In many applications, however, the graph that defines which vertex sets are feasible is not the same as the graph that governs how tokens, agents, or resources may move. Existing models such as token sliding and token jumping typically do not distinguish the problem graph and the movement graph. Motivated by this mismatch, we introduce a directed weighted two-graph model that cleanly separates feasibility from movement. A problem graph specifies the desired combinatorial objects, while a movement graph specifies admissible relocations and their costs. This yields a flexible framework that captures asymmetry, heterogeneous movement constraints, and weighted transitions, while subsuming classical discovery models as special cases. We investigate this model through \textsc{Path Discovery} and \textsc{Shortest Path Discovery}, where the task is to realize a vertex set containing an $s$-$t$-path or a shortest $s$-$t$-path in the problem graph. These problems are particularly natural in applications, since directed and weighted shortest paths are among the most fundamental algorithmic primitives. At the same time, previous work has already shown that discovery can be computationally hard even when the underlying optimization problem is easy. Our results show that this phenomenon persists, and becomes especially rich, in the two-graph setting. We obtain a detailed complexity picture, identifying tractable cases as well as strong hardness results.

翻译：我们研究解决方案发现，其目标是通过一系列有界局部移动，从初始配置获得问题的可行解。然而，在许多应用中，定义可行顶点集的图与控制标记、智能体或资源如何移动的图并不相同。现有模型（如令牌滑动和令牌跳跃）通常不区分问题图与移动图。受此错配启发，我们引入一个有向加权双图模型，将可行性分析与移动性清晰分离：问题图指定所需的组合对象，而移动图规定可允许的重新定位及其代价。这一框架具备灵活性，能够刻画非对称性、异构移动约束以及加权转移，同时将经典发现模型作为特例纳入其中。我们通过路径发现和最短路径发现问题对该模型进行研究，其目标是在问题图中实现包含一条s-t路径或最短s-t路径的顶点集。这些问题在应用中非常自然，因为带权有向最短路径是最基础的算法原语之一。与此同时，已有研究表明，即便底层优化问题简单，解决方案发现仍可能具有计算难度。我们的结果表明，在双图设置下，这一现象持续存在且变得尤为丰富。我们获得了详细的复杂度刻画图谱，既识别了可处理情形，也揭示了强难解性结果。