AI 中文总结
本文证明无三角形图上最大独立集的n^{1/2-ε}近似问题是NP难的，通过推广归约方案得出排除含环子图的图族的相关不可近似性结论。
AI 中文摘要
对于任意ε>0，在n顶点图中近似最大独立集（Max Independent Set）达到n^{1-ε}是NP难问题[Hastad '96, Zuckerman '07]。在无三角形图中，一个简单的论证给出了多项式时间的n^{1/2}近似算法；而对于任意ε>0，若存在n^{1/4-ε}近似算法，则将推出NP⊆BPP[Bonnet, Thomassé, Tran, Watrigant; ESA '20]。本文中，我们通过证明对n^{1/2-ε}近似算法的对应难解性来闭合这一差距。归约过程非常简单，使用Moser-Tardos重抽样算法使构造的图无三角形。可靠性分析使用了Haeupler、Saha和Srinivasan基于Moser与Tardos证明的结果，以在Moser-Tardos算法终止后，对一个固定的相对较大子集是独立集的概率进行上界估计。我们将该方案推广，证明对于任意非空有限图族ℱ，其中每个图都包含至少一个环，对于任意ε>0，若存在排除ℱ中每个成员作为子图的图上最大独立集的n^{μ(ℱ)-ε}近似算法，则将推出NP⊆BPP，其中μ(ℱ):=1 - max_{H∈ℱ} min_{U⊆V(H), H[U]包含一个环} (|U|-2)/(|E(H[U])|-1)。
英文摘要
For every $\varepsilon > 0$, it is NP-hard to $n^{1-\varepsilon}$-approximate Max Independent Set in $n$-vertex graphs [Hastad '96, Zuckerman '07]. In triangle-free graphs, a simple argument gives a polynomial-time $n^{1/2}$-approximation algorithm, whereas, for every $\varepsilon > 0$, an $n^{1/4-\varepsilon}$-approximation algorithm would imply that NP $\subseteq$ BPP [Bonnet, Thomassé, Tran, Watrigant; ESA '20]. In this note, we close this gap by proving the corresponding hardness against $n^{1/2-\varepsilon}$-approximation algorithms. The reduction is very simple and uses the Moser-Tardos resampling algorithm to make the constructed graphs triangle-free. The soundness uses a result of Haeupler, Saha, and Srinivasan building on the proof of Moser and Tardos, to upper-bound the probability that a fixed relatively large subset is an independent set after the Moser-Tardos algorithm terminates. We generalize this scheme and show that, for any nonempty finite family $\mathcal F$ of graphs, each containing at least one cycle, for any $\varepsilon > 0$, an $n^{μ(\mathcal F)-\varepsilon}$-approximation algorithm for Max Independent Set in graphs excluding every member of $\mathcal F$ as a subgraph implies that NP $\subseteq$ BPP, where $μ(\mathcal F) := 1 - \max\limits_{H \in \mathcal F}~\min\limits_{U \subseteq V(H), H[U] \text{contains a cycle}} (|U|-2)/(|E(H[U])|-1)$.
Comments11 pages, 1 figure
