In the recently introduced framework of solution discovery via reconfiguration [Fellows et al., ECAI 2023], we are given an initial configuration of $k$ tokens on a graph and the question is whether we can transform this configuration into a feasible solution (for some problem) via a bounded number $b$ of small modification steps. In this work, we study solution discovery variants of polynomial-time solvable problems, namely Spanning Tree Discovery, Shortest Path Discovery, Matching Discovery, and Vertex/Edge Cut Discovery in the unrestricted token addition/removal model, the token jumping model, and the token sliding model. In the unrestricted token addition/removal model, we show that all four discovery variants remain in P. For the toking jumping model we also prove containment in P, except for Vertex/Edge Cut Discovery, for which we prove NP-completeness. Finally, in the token sliding model, almost all considered problems become NP-complete, the exception being Spanning Tree Discovery, which remains polynomial-time solvable. We then study the parameterized complexity of the NP-complete problems and provide a full classification of tractability with respect to the parameters solution size (number of tokens) $k$ and transformation budget (number of steps) $b$. Along the way, we observe strong connections between the solution discovery variants of our base problems and their (weighted) rainbow variants as well as their red-blue variants with cardinality constraints.

翻译：在最近提出的通过重配置进行解发现的框架中[Fellows等人，ECAI 2023]，给定图上的$k$个令牌的初始配置，问题在于是否可以通过有界数量$b$的小修改步骤将此配置转换为可行解（针对某些问题）。在本工作中，我们研究了多项式时间可解问题的解发现变体，即在无限制令牌添加/移除模型、令牌跳跃模型和令牌滑动模型中的生成树发现、最短路径发现、匹配发现以及顶点/边割发现。在无限制令牌添加/移除模型中，我们证明所有四个发现变体仍属于P类。对于令牌跳跃模型，我们同样证明了包含于P类，但顶点/边割发现除外，我们证明了其为NP完全问题。最后，在令牌滑动模型中，几乎所有考虑的问题都变为NP完全问题，但生成树发现是个例外，它仍可在多项式时间内求解。随后，我们研究了NP完全问题的参数化复杂性，并根据参数解规模（令牌数量）$k$和转换预算（步骤数量）$b$提供了可处理性的完整分类。在此过程中，我们观察到基础问题的解发现变体与其（加权）彩虹变体以及具有基数约束的红蓝变体之间存在强关联。