利用图结构实现近最优广播
Exploiting Graph Structure for Near-Optimal Broadcasting
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中文总结 AI 辅助
研究利用图结构实现近最优广播，改进精确算法为近似算法，给出有界顶点完整性图上的近似算法，证明部分参数化硬度，还给出距离团、距离路径及极图的相关算法。
中文摘要 AI 辅助
电话广播是在网络中传播信息的经典模型。给定一个带源顶点s的连通图G(V,E)，每个已通知顶点在每个时间步可通知恰好一个未通知邻居。广播问题是问能否在t步内通知所有顶点，最小的t值即广播时间b(G,s)，相关变体考虑最坏情况源b(G)。这两个变体都是NP难的，且每个n顶点图满足b(G,s)≥log₂n。Fomin等人最近给出了基于图结构参数的FPT算法。本文研究更快的近似算法，改进了Fomin等人的O*(3ⁿ)精确算法为O*((3 - f(x))ⁿ)算法并带有+x加法近似，还给出了有界顶点完整性图上的近似算法。同时证明了顶点覆盖高于最大匹配、支配集大小和图直径的参数化硬度。最后给出了距离团的+2加法近似算法、距离路径的2因子近似算法和极图的多项式时间算法。
英文摘要
Telephone broadcasting is a classical model for spreading information in a network. Given a connected graph $G(V,E)$ with source vertex $s$, each informed vertex may inform exactly one uninformed neighbor in every time step. The \textsc{Broadcasting} problem asks whether all vertices can be informed within $t$ steps; the minimum such value is the broadcast time $b(G,s)$. A related variant considers the worst-case source, $b(G)=\max_{u\in V} b(G,u)$. Both variants are NP-hard, and every $n$-vertex graph satisfies $b(G,s)\ge \log_2 n$. Fomin \textit{et al.}~\cite{fomin2023parameterized} recently gave FPT algorithms for this problem under several structural graph parameters. Instead of computing optimal broadcast schedules, we study faster approximation algorithms that produce valid schedules. We improve the $O^*(3^n)$ exact algorithm of Fomin \textit{et al.} to an $O^*((3-f(x))^n)$ algorithm with a $+x$ additive approximation, where $f(x)>0$ is a constant for every fixed $x$. We also give approximation algorithms on graphs of bounded vertex integrity, including a polynomial-time $+2k$ additive approximation algorithm. Complementing these positive results, we prove parameterized hardness for vertex cover above maximum matching ($\mathrm{VC}-\mathrm{MM}$), dominating set size, and graph diameter, indicating that FPT algorithms for these parameters are unlikely. Finally, we present a $+2$ additive approximation algorithm for distance-to-clique running in $O^*(2^{O(k\log k)})$ time, a $2$-factor approximation algorithm for distance-to-path running in XP time, and a polynomial-time algorithm for polar graphs.
