[论文解读] Uniform Kernelization Complexity of Hitting Forbidden Minors

Archontia C. Giannopoulou, Bart M. P. Jansen+2 more|arXiv (Cornell University)|Feb 13, 2015

Advanced Graph Theory Research参考文献 12被引用 8

一句话总结

本文通过证明：尽管树宽-$\eta$ 删除问题不承认统一多项式内核（除非 NP $\subseteq$ coNP/poly），但树深度-$\eta$ 删除问题确实承认大小为 $\mathcal{O}(k^6)$ 的统一多项式内核，该结果通过深入分析树深度分解的显式结构，并采用非凸包基内核化算法实现。

ABSTRACT

The F-Minor-Free Deletion problem asks, for a fixed set F and an input consisting of a graph G and integer k, whether k vertices can be removed from G such that the resulting graph does not contain any member of F as a minor. This paper analyzes to what extent provably effective and efficient preprocessing is possible for F-Minor-Free Deletion. Fomin et al. (FOCS 2012) showed that the special case Planar F-Deletion (when F contains at least one planar graph) has a kernel of size f(F) * k^{g(F)} for some functions f and g. The degree g of the polynomial grows very quickly; it is not even known to be computable. Fomin et al. left open whether Planar F-Deletion has kernels whose size is uniformly polynomial, i.e., of the form f(F) * k^c for some universal constant c that does not depend on F. Our results in this paper are twofold. (1) We prove that some Planar F-Deletion problems do not have uniformly polynomial kernels (unless NP is in coNP/poly). In particular, we prove that Treewidth-Eta Deletion does not have a kernel with O(k^{eta/4} - eps) vertices for any eps > 0, unless NP is in coNP/poly. In fact, we even prove the kernelization lower bound for the larger parameter vertex cover number. This resolves an open problem of Cygan et al. (IPEC 2011). It is a natural question whether further restrictions on F lead to uniformly polynomial kernels. However, we prove that even when F contains a path, the degree of the polynomial must, in general, depend on the set F. (2) A canonical F-Minor-Free Deletion problem when F contains a path is Treedepth-eta Deletion: can k vertices be removed to obtain a graph of treedepth at most eta? We prove that Treedepth-eta Deletion admits uniformly polynomial kernels with O(k^6) vertices for every fixed eta. In order to develop the kernelization we prove several new results about the structure of optimal treedepth-decompositions.

研究动机与目标

• 确定所有平面 $\mathcal{F}$-无小器删除问题是否都承认统一多项式内核。
• 研究在内核化背景下，树宽与树深度参数之间的结构性差异。
• 解决树宽-$\eta$ 删除问题内核化复杂度的一个开放问题。
• 为树深度-$\eta$ 删除问题开发一种构造性、非凸包基的内核化算法。

提出的方法

• 通过从已知困难性假设约化，证明树宽-$\eta$ 删除问题不承认统一多项式内核。
• 基于最优树深度分解的结构性质，提出一种新颖的树深度-$\eta$ 删除问题内核化框架。
• 采用有界分支约化策略，通过分解树的高度与叶结构，控制内核中连通分量的大小。
• 通过结合顶点覆盖、反馈顶点集和连通分量树的界，建立内核大小为 $\mathcal{O}(k^6)$ 的上界。
• 通过构建完全显式的内核化算法，避免依赖于“凸包机械装置”。
• 利用障碍集与内核大小之间的联系，推导出内核复杂度的上下界。

实验结果

研究问题

• RQ1 每个平面 $\mathcal{F}$-无小器删除问题是否都承认统一多项式内核，即大小为 $f(\mathcal{F}) \cdot k^c$，其中 $c$ 为通用常数？
• RQ2 在无小器删除问题中，树宽与树深度参数的内核化复杂度是否存在根本性差异？
• RQ3 能否通过一种构造性、非凸包基的内核化算法，为树深度-$\eta$ 删除问题实现统一多项式内核大小？
• RQ4 在无小器闭合图族中，障碍集大小与内核大小之间存在何种关系？

主要发现

• 除非 NP $\subseteq$ coNP/poly，否则树宽-$\eta$ 删除问题不承认大小为 $\mathcal{O}(k^{\eta/4 - \epsilon})$ 的内核，其中 $\epsilon > 0$。
• 即使参数化为顶点覆盖数，树宽-$\eta$ 删除问题在相同假设下仍不承认统一多项式内核。
• 树深度-$\eta$ 删除问题承认大小为 $\mathcal{O}(k^6)$ 的统一多项式内核，显式上界为 $2^{\mathcal{O}(\eta^2)} \cdot k^6$。
• 该内核化算法完全构造性，不依赖于先前工作中使用的非构造性“凸包机械装置”。
• 通过 $k$ 次删除可归约至树深度 $\eta$ 的图的障碍集大小被限制在 $2^{\mathcal{O}(\eta^2)} \cdot k^6$ 以内，与内核大小上界一致。
• 研究结果在无小器闭合图族中建立了内核化复杂度与障碍集大小之间的紧密联系。