[论文解读] Polynomial Pass Semi-Streaming Lower Bounds for K-Cores and Degeneracy

Sepehr Assadi, Prantar Ghosh+3 more|arXiv (Cornell University)|Jan 1, 2024

Complexity and Algorithms in Graphs被引用 2

一句话总结

本文首次建立了k-cores和退化性问题在半流模型中的多项式通过数下界，证明任何需要精确计算的半流算法必须进行Ω(n^{1/3})轮处理。作者提出一种基于广义隐藏指针追踪问题（MultiHPC）的新通信协议，实现近乎线性的通信开销，并证明了最优的轮次-通信复杂度下界，将此前关于精确解的下界从n^{1/5}显著提升至n^{1/3}}轮处理。

ABSTRACT

The following question arises naturally in the study of graph streaming algorithms: Is there any graph problem which is "not too hard", in that it can be solved efficiently with total communication (nearly) linear in the number n of vertices, and for which, nonetheless, any streaming algorithm with Õ(n) space (i.e., a semi-streaming algorithm) needs a polynomial n^Ω(1) number of passes? Assadi, Chen, and Khanna [STOC 2019] were the first to prove that this is indeed the case. However, the lower bounds that they obtained are for rather non-standard graph problems. Our first main contribution is to present the first polynomial-pass lower bounds for natural "not too hard" graph problems studied previously in the streaming model: k-cores and degeneracy. We devise a novel communication protocol for both problems with near-linear communication, thus showing that k-cores and degeneracy are natural examples of "not too hard" problems. Indeed, previous work have developed single-pass semi-streaming algorithms for approximating these problems. In contrast, we prove that any semi-streaming algorithm for exactly solving these problems requires (almost) Ω(n^{1/3}) passes. The lower bound follows by a reduction from a generalization of the hidden pointer chasing (HPC) problem of Assadi, Chen, and Khanna, which is also the basis of their earlier semi-streaming lower bounds. Our second main contribution is improved round-communication lower bounds for the underlying communication problems at the basis of these reductions: - We improve the previous lower bound of Assadi, Chen, and Khanna for HPC to achieve optimal bounds for this problem. - We further observe that all current reductions from HPC can also work with a generalized version of this problem that we call MultiHPC, and prove an even stronger and optimal lower bound for this generalization. These two results collectively allow us to improve the resulting pass lower bounds for semi-streaming algorithms by a polynomial factor, namely, from n^{1/5} to n^{1/3} passes.

研究动机与目标

• 通过证明k-cores和退化性等自然的、‘不太难’的问题在半流模型中需要多项式轮数处理，填补图流理论中的空白。
• 展示尽管存在高效的单轮近似算法，k-cores和退化性的精确计算仍无法在亚多项式轮数内完成。
• 为退化性问题设计一种具有近乎线性通信开销的新通信协议，表明这些问题天然适合作为强下界候选。
• 改进隐藏指针追踪（HPC）问题及其推广（MultiHPC）的轮次-通信复杂度下界，实现最优性。

提出的方法

• 作者引入一种广义通信问题，称为MultiHPC（多层隐藏指针追踪），扩展原始HPC框架，以支持更强的归约。
• 证明：若MultiHPC存在短通信协议，则集交问题存在低信息量的ε-求解器，进而可推出精确求解器，基于信息论论证。
• 提出一种新颖的构件构造方法，将MultiHPC归约至退化性问题，将通信复杂度嵌入图流实例中。
• 利用该归约将任何退化性问题的半流算法转化为通信协议，建立流处理轮数复杂度与通信轮数复杂度之间的联系。
• 通过正三角差异和基于熵的不等式，建立信息复杂度的紧致界。
• 构造出一种通信开销为eO(n)的退化性问题通信协议，表明该问题在通信复杂度上‘不太难’，从而强化了下界结果的重要性。

实验结果

研究问题

• RQ1 能否在半流模型中为k-cores和退化性等自然图问题建立多项式通过数下界？
• RQ2 是否存在一种通信问题，既能捕捉退化性和k-cores的复杂度，又允许近乎线性的通信开销？
• RQ3 能否将隐藏指针追踪（HPC）问题推广，以实现最优的通信复杂度下界？
• RQ4 任何半流算法精确计算退化性或k-cores所需的最优通过数是多少？
• RQ5 HPC及其推广在通信复杂度下界上的改进，如何转化为更强的流处理轮数下界？

主要发现

• 本文证明，任何用于精确计算退化性或k-cores的半流算法都必须进行Ω(n^{1/3})轮处理，优于此前的n^{1/5}下界。
• 构造了一种通信开销为eO(n)的退化性问题新通信协议，表明该问题在通信复杂度上‘不太难’。
• 作者提出并分析了MultiHPC问题，即隐藏指针追踪的推广版本，支持更强的归约并实现最优下界。
• HPC的轮次-通信复杂度得到最优界，MultiHPC亦同理，填补了先前研究的空白。
• 从MultiHPC到退化性的归约表明，任何高效的退化性半流算法都将意味着存在一个用于MultiHPC的短通信协议，而根据推导出的下界，这是不可能的。
• 综合来看，结果表明k-cores和退化性是自然的例子：它们在近乎线性通信下可高效求解，但在半流模型中却需要多项式轮数处理。