AI 中文总结
针对时间网络层级发现的Seg-Agony问题，研究其参数化复杂度，明确了不同等级数、惩罚值等参数下的复杂度边界与可解性结果。
AI 中文摘要
现实世界的网络通常以多层结构组织，形成决定各独立组件间交互方式的层级体系。为了在时间网络中发现此类层级结构，Tatti[ECML PKDD 2018]提出了时间痛苦值问题Seg-Agony。该问题的目标是为每个顶点分配一个（从1到$k$的）特定等级，使得弧仅从较低等级指向较高等级。反向弧会根据对应等级的差值受到惩罚。由于弧可能随时间变化，允许每个顶点改变其等级$\nell\ne 1$次，以最小化总惩罚值$\nalpha$（称为时间痛苦值）。\n我们研究了Seg-Agony的参数化复杂度，重点关注可能的等级数量$k$，并确定了精确的复杂度边界。我们证明，当$k=2$时，该问题可在多项式时间内求解；当$k=3$且$\nell=1$时，该问题是NP难的，但当$\nalpha$为常数时可在多项式时间内求解；当$k=4$且$\nell=1$时，即使$\nalpha=0$，该问题仍是NP难的。我们进一步提出了一种适用于顶点数$n$为常数情况的多项式时间算法，并证明了该问题关于组合参数$n+\nell$是固定参数可处理的。
英文摘要
Real-world networks are often organized in several layers forming a hierarchy which determines the interaction between the individual components. In order to discover such hierarchies in temporal networks, Tatti [ECML PKDD 2018] introduced the temporal agony problem Seg-Agony. Here, the goal is to assign each vertex a certain rank (from 1 to $k$) such that arcs only point from lower ranks to higher ranks. Backward arcs are penalized depending on the difference between the corresponding ranks. Since arcs may change over time, each vertex is allowed to change its rank $\ell\ge 1$ times in order to minimize the overall penalty $α$ (called temporal agony). We study the parameterized complexity of Seg-Agony with a special focus on the number $k$ of possible ranks for which we identify the precise complexity border. We show that the problem is polynomial-time solvable for $k=2$, NP-hard for $k=3$ and $\ell=1$ but polynomial-time solvable for constant $α$, and NP-hard for $k=4$ and $\ell=1$ even for $α=0$. We further show a polynomial-time algorithm for a constant number $n$ of vertices and fixed-parameter tractability for the combined parameter $n+\ell$.
CommentsTo appear at ALGOWIN 2026
