AI 中文总结
该研究针对容错数据结构的替换路径覆盖问题，提出两个简洁构造，改进了上下界，得到f=O(L)范围下的近紧覆盖值
AI 中文摘要
设L和f为正整数，图G的一个(L,f)-替换路径覆盖（RPC）是子图族G，使得对任意最多含f条边的集合F，存在子族G_F⊆G满足：（1）G_F中所有子图均不含F中的边；（2）对每对顶点s、t，若G-F中存在长度不超过L条边的最短路径，则该路径也存在于G_F的某个子图中。子图总数|G|称为覆盖值。RPC是容错数据结构设计的重要工具。Weimann与Yuster[TALG 2013]提出覆盖值为Õ(f L^f)的RPC；Karthik与Parter[TALG 2024]证明至少需要Ω((L/f)^f)个子图；近期Bilò、Chechik、Choudhary、Cohen与Schirneck[ICALP 2026]针对极小灵敏度f=o(log L)设计了新方法，覆盖值为Õ(f e^f (L/f)^{f+o(1)})，同时证明互补范围f=Ω(log L)的RPC必须含Ω((√(f e^f)/L)·(L/f)^f)个子图，这留下了真实覆盖值的问题。我们给出两个极为简洁的构造，同时改进上下界，得到更宽范围f=O(L)下的近紧覆盖值Õ( ((L+f)^{L+f})/(L^L f^f) )·poly(f)
英文摘要
Let $L$ and $f$ be positive integers. An $(L,f)$-replacement path covering (RPC) for a graph $G$ is a family $\mathcal{G}$ of subgraphs such that, for every set $F$ of at most $f$ edges, there is a subfamily $\mathcal{G}_F \subseteq \mathcal{G}$ with the following properties. (1) No subgraph in $\mathcal{G}_F$ contains an edge of $F$. (2) For each pair of vertices $s,t$ that have a shortest path in $G{-}F$ with at most $L$ edges, one such path also exists in some subgraph in $\mathcal{G}_F$. The total number $|\mathcal{G}|$ of subgraphs is called the covering value. RPCs are an important tools in the design of fault-tolerant data structures. Weimann and Yuster [TALG 2013] presented an RPC with covering value $\widetilde{O}(f L^f)$. Karthik and Parter [TALG 2024] showed that $Ω( (L/f)^f )$ subgraphs are necessary. Recently, Bilò, Chechik, Choudhary, Cohen, and Schirneck [ICALP 2026] devised a new approach for very small sensitivities $f = o(\log L)$ with covering value $\widetilde{O}(f e^f (L/f)^{f+o(1)})$. They also showed that any RPC in the complementary range $f = Ω(\log L)$ must contain $Ω( (\sqrt{f e^f}/L) \cdot (L/f)^f)$ subgraphs. This left open the question of what is the true covering value. We give two surprisingly simple constructions that improve both the upper and lower bound. This results in a near-tight covering value of $\widetildeΘ(\frac{(L+f)^{L+f}}{L^L f^f}) \cdot \mathsf{poly}(f)$ for the much wider range of $f = O(L)$.
