We study the kernelization of exploration problems on temporal graphs. A temporal graph consists of a finite sequence of snapshot graphs $\mathcal{G}=(G_1, G_2, \dots, G_L)$ that share a common vertex set but might have different edge sets. The non-strict temporal exploration problem (NS-TEXP for short) introduced by Erlebach and Spooner, asks if a single agent can visit all vertices of a given temporal graph where the edges traversed by the agent are present in non-strict monotonous time steps, i.e., the agent can move along the edges of a snapshot graph with infinite speed. The exploration must at the latest be completed in the last snapshot graph. The optimization variant of this problem is the $k$-arb NS-TEXP problem, where the agent's task is to visit at least $k$ vertices of the temporal graph. We show that under standard computational complexity assumptions, neither of the problems NS-TEXP nor $k$-arb NS-TEXP allow for polynomial kernels in the standard parameters: number of vertices $n$, lifetime $L$, number of vertices to visit $k$, and maximal number of connected components per time step $\gamma$; as well as in the combined parameters $L+k$, $L + \gamma$, and $k+\gamma$. On the way to establishing these lower bounds, we answer a couple of questions left open by Erlebach and Spooner. We also initiate the study of structural kernelization by identifying a new parameter of a temporal graph $p(\mathcal{G}) = \sum_{i=1}^{L} (|E(G_i)|) - |V(G)| +1$. Informally, this parameter measures how dynamic the temporal graph is. Our main algorithmic result is the construction of a polynomial (in $p(\mathcal{G})$) kernel for the more general Weighted $k$-arb NS-TEXP problem, where weights are assigned to the vertices and the task is to find a temporal walk of weight at least $k$.

翻译：我们研究了时间图上的探索问题的核化。时间图由有限个快照图序列 $\mathcal{G}=(G_1, G_2, \dots, G_L)$ 组成，这些快照图共享同一顶点集但可能具有不同的边集。Erlebach 和 Spooner 提出的非严格时间探索问题（简称 NS-TEXP）询问：单个智能体能否访问给定时间图的所有顶点，其中智能体遍历的边出现在非严格单调的时间步中，即智能体可以以无限速度沿快照图的边移动。探索最迟必须在最后一个快照图中完成。该问题的优化变体是 $k$-arb NS-TEXP 问题，其中智能体的任务是访问时间图中至少 $k$ 个顶点。我们证明，在标准计算复杂性假设下，NS-TEXP 和 $k$-arb NS-TEXP 问题在标准参数（顶点数 $n$、生命周期 $L$、需访问顶点数 $k$ 以及每时间步最大连通分量数 $\gamma$）以及组合参数 $L+k$、$L + \gamma$ 和 $k+\gamma$ 上均不允许多项式核。在建立这些下界的过程中，我们回答了 Erlebach 和 Spooner 留下的若干未解问题。我们还通过识别时间图的新参数 $p(\mathcal{G}) = \sum_{i=1}^{L} (|E(G_i)|) - |V(G)| +1$ 开创了结构核化的研究。直观上，该参数衡量时间图的动态程度。我们的主要算法结果是为更通用的加权 $k$-arb NS-TEXP 问题构建了一个关于 $p(\mathcal{G})$ 的多项式核，其中顶点被赋予权重，任务是在时间图中找到权重大于等于 $k$ 的游走。