发表机构
University of Cambridge; University of Bristol（剑桥大学; 布里斯托大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对单遍边到达流中的最大团和最大独立集问题，证明了任何确定性算法需 Ω(n^2/(β log n)) 位空间，填补了与随机算法的差距，凸显随机性的优势。
AI 中文摘要
我们研究单遍边到达图流设置中的经典最大团和最大独立集问题。在此设置中，输入图 G = (V,E) 的边逐一呈现（可能包括删除），之后算法需要在流结束时产生一个大的团或独立集，重点在于空间复杂度。我们感兴趣的是为任意 β ≥ 1 寻找 β 近似解。先前的工作给出了一种使用 O~(n^2/β^2) 位空间的算法，以及相应的 Ω~(n^2/β^2) 双方通信下界 [Halldórsson 等人，ICALP'12]，看似解决了该问题。然而，他们的算法关键依赖于随机性，而目前已知的最佳确定性算法仍是使用 O(n^2/β) 位空间的经典去随机化方法，留下了大小为 O~(β) 的（确定性）差距。我们用一个（几乎）紧的下界解决了这个确定性差距：任何确定性算法解决这两个问题中的任何一个都必须使用 Ω(n^2/(β·log n)) 位空间。我们的证明通过双方单向通信下界实现，并凸显了在处理这两个问题时随机性的力量。
英文摘要
We study the canonical \textsf{Maximum Clique} and \textsf{Maximum Independent Set} problems in the one-pass edge-arrival graph streaming setting. Here, the edges of some input graph $G = (V,E)$ are presented one at a time (possibly including deletions), before an algorithm needs to produce either a large clique or independent set at the end of the stream, with the focus being on space complexity. We are interested in finding $β$-approximate solutions, for any $β\geq 1$. Previous work gave an algorithm using $\tilde{O}\left(n^2/β^2\right)$ bits of space, together with a corresponding $\tildeΩ\left(n^2/β^2\right)$ two-party communication lower bound [Halldórsson et al., ICALP'12], seeming to resolve the problem. However, their algorithm crucially relies on randomness, and the best known deterministic algorithm remains a folklore derandomisation using $O\left(n^2/β\right)$ bits of space, leaving a (deterministic) gap of size $\tilde{O}(β)$. We resolve this deterministic gap with an (almost) tight lower bound: any deterministic algorithm for either problem must use $Ω\left(\frac{n^2}{β\cdot\log n}\right)$ bits of space. Our proof is via a two-party one-way communication lower bound, and highlights the power of randomness when approaching either of these problems.
Comments24 pages, 1 figure
