Sub-polynomial parameterized complexity of $k$-core

2026-09-21

AI Summary

研究通过图的treewidth、k值及特定类型图参数化方法，解决k-core问题在次多项式复杂度类中的定位，提出并证明了若干算法及其复杂度界限。

Abstract

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.

Problem

• k-core
• parameterized complexity
• parallel algorithms
• treewidth
• chordal graphs

Innovation

• sub-polynomial complexity
• parameterized by treewidth
• chordal graphs
• interval graphs
• pathwidth