发表机构
LI-PARAD, Université de Versailles Saint-Quentin-en-Yvelines, Université Paris-Saclay（凡尔赛圣康坦伊夫林大学，巴黎萨克雷大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对链接流最大团枚举问题，基于瞬时图的最大度Δ和简并度k给出紧界，提出两种多项式延迟算法及在线版本，还将结果拓展至时间图的特定团类型。
AI 中文摘要
链接流是顶点对之间的交互集合，每个交互持续某一时间段；链接流的团是一组顶点及一个时间段，该时间段内所有顶点均发生交互。若团的顶点集或时间段均无法再扩大，则称其为最大团。所有已知的枚举链接流最大团的算法，对每个输出的团，其运行时间可能随团的大小呈指数增长。本文基于链接流瞬时图的两个参数——最大度Δ和简并度k——开展研究，证明含m条边的链接流有O(mΔ²3^(Δ/3))个最大团，含n个顶点且有|T|个不同结束时间的链接流有O(n|T|k²3^(k/3))个最大团；这两个界分别在Δ和k的多项式因子内是紧的，且第二个界将此前界中的2^k因子改进为3^(k/3)。随后本文提出两种算法：第一种算法的建立时间为O(m log m)，多项式延迟为poly(Δ)log m；第二种算法的建立时间为O(m log m + mk² log³ n)，多项式延迟为poly(k)log m。据本文所知，这是该问题首批具有多项式延迟的算法。本文还给出第二种算法的在线版本，并证明所得结果适用于时间图的Δ-团和(Δ,γ)-团。
英文摘要
A link stream is a set of interactions between pairs of vertices, each one lasting during some time interval, and a clique of a link stream is a set of vertices together with a time interval during which all of them interact. A clique is maximal if neither its vertex set nor its interval can be enlarged. All known algorithms listing the maximal cliques of a link stream may spend time exponential in the size of a clique for each clique they output. Here, we approach the question under the light of two parameters of the instantaneous graphs of the stream, their maximum degree $Δ$ and their degeneracy $k$. We prove that a link stream with $m$ links has $O(mΔ^2 3^{Δ/3})$ maximal cliques, and that a link stream with $n$ vertices and $|\mathcal{T}|$ distinct end times has $O(n|\mathcal{T}|k^2 3^{k/3})$ maximal cliques. Both bounds are tight up to a factor polynomial in $Δ$ and $k$ respectively, and the second one improves the factor $2^k$ of previous bounds to $3^{k/3}$. Then we present two algorithms. The first one has setup time $O(m\log m)$ and polynomial time delay $\mathrm{poly}(Δ)\log m$. The second one has setup time $O(m\log m+mk^2\log^3 n)$ and polynomial time delay $\mathrm{poly}(k)\log m$. To the best of our knowledge, these are the first algorithms with polynomial time delay for this problem. We also give an online version of the second algorithm, and we show that our results apply to the $Δ$-cliques and $(Δ,γ)$-cliques of temporal graphs.
