$k$-核的次多项式参数化复杂度

中文摘要 AI 辅助

图的$k$-核是其（唯一的）最小度至少为$k$的最大子图。对于任意$k \geq 3$，判定给定顶点是否属于$k$-核是一个P完全问题，这意味着该问题本质上是顺序的，即使在最大度为$k+1$的图上，也极不可能存在高效的并行算法。本文研究$k$-核问题的替代参数化，以确定在何种条件下该问题可归入次多项式复杂度类。我们证明，当以树宽为参数时，该问题属于para-NC$^{2+\epsilon}$；当以$k$为参数且图是弦图时，该问题属于para-NC$^3$。此外，当$k = \mathcal{O}(\lg v(G))$时，我们为区间图提出了一种新的NC$^{3}$算法，该算法依赖于对路径宽度的改进参数化。最后，我们建立了相应的下界，证明即使采用这些参数化，计算$k$-核仍然是L困难的，即至少需要对数空间。这些发现通过突出使$k$-核问题本质上顺序化的图参数，探索了该问题并行可处理性的边界。

英文摘要

The $k$-core of a graph is its (unique) largest subgraph with minimum degree at least $k$. For any $k \geq 3$, deciding whether a given vertex belongs to the $k$-core is a P-complete problem, meaning that it is inherently sequential and highly unlikely to admit efficient parallel algorithms, even on graphs of maximum degree $k+1$. This paper investigates alternative parameterizations of the $k$-core problem to identify conditions under which it can be placed into sub-polynomial complexity classes. We prove that the problem is in para-NC$^{2+ε}$ when parameterized by treewidth, and in para-NC$^3$ when parameterized by $k$ on chordal graphs. Furthermore, we introduce a novel NC$^{3}$ algorithm for interval graphs when $k = \mathcal{O}(\lg v(G))$, which relies on an improved parameterization by pathwidth. Finally, we establish corresponding lower bounds, demonstrating that, even with these parameterizations, computing the $k$-core remains L-hard, meaning it requires at least logarithmic space. These findings explore the boundary of parallel tractability for the $k$-core problem by highlighting the graph parameters that make it inherently sequential.

发表机构

• University of São Paulo（圣保罗大学）

机构由 AI 辅助整理，请以论文原文为准。