Parameterized Complexity of Streaming Diameter and Connectivity Problems

Jelle J. Oostveen, E. J. V. Leeuwen
{"title":"Parameterized Complexity of Streaming Diameter and Connectivity Problems","authors":"Jelle J. Oostveen, E. J. V. Leeuwen","doi":"10.48550/arXiv.2207.04872","DOIUrl":null,"url":null,"abstract":"We initiate the investigation of the parameterized complexity of Diameter and Connectivity in the streaming paradigm. On the positive end, we show that knowing a vertex cover of size $k$ allows for algorithms in the Adjacency List (AL) streaming model whose number of passes is constant and memory is $O(\\log n)$ for any fixed $k$. Underlying these algorithms is a method to execute a breadth-first search in $O(k)$ passes and $O(k \\log n)$ bits of memory. On the negative end, we show that many other parameters lead to lower bounds in the AL model, where $\\Omega(n/p)$ bits of memory is needed for any $p$-pass algorithm even for constant parameter values. In particular, this holds for graphs with a known modulator (deletion set) of constant size to a graph that has no induced subgraph isomorphic to a fixed graph $H$, for most $H$. For some cases, we can also show one-pass, $\\Omega(n \\log n)$ bits of memory lower bounds. We also prove a much stronger $\\Omega(n^2/p)$ lower bound for Diameter on bipartite graphs. Finally, using the insights we developed into streaming parameterized graph exploration algorithms, we show a new streaming kernelization algorithm for computing a vertex cover of size $k$. This yields a kernel of $2k$ vertices (with $O(k^2)$ edges) produced as a stream in $\\text{poly}(k)$ passes and only $O(k \\log n)$ bits of memory.","PeriodicalId":137775,"journal":{"name":"International Symposium on Parameterized and Exact Computation","volume":"17 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2022-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Symposium on Parameterized and Exact Computation","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.48550/arXiv.2207.04872","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

Abstract

We initiate the investigation of the parameterized complexity of Diameter and Connectivity in the streaming paradigm. On the positive end, we show that knowing a vertex cover of size $k$ allows for algorithms in the Adjacency List (AL) streaming model whose number of passes is constant and memory is $O(\log n)$ for any fixed $k$. Underlying these algorithms is a method to execute a breadth-first search in $O(k)$ passes and $O(k \log n)$ bits of memory. On the negative end, we show that many other parameters lead to lower bounds in the AL model, where $\Omega(n/p)$ bits of memory is needed for any $p$-pass algorithm even for constant parameter values. In particular, this holds for graphs with a known modulator (deletion set) of constant size to a graph that has no induced subgraph isomorphic to a fixed graph $H$, for most $H$. For some cases, we can also show one-pass, $\Omega(n \log n)$ bits of memory lower bounds. We also prove a much stronger $\Omega(n^2/p)$ lower bound for Diameter on bipartite graphs. Finally, using the insights we developed into streaming parameterized graph exploration algorithms, we show a new streaming kernelization algorithm for computing a vertex cover of size $k$. This yields a kernel of $2k$ vertices (with $O(k^2)$ edges) produced as a stream in $\text{poly}(k)$ passes and only $O(k \log n)$ bits of memory.
流直径的参数化复杂度与连通性问题
我们对流范式中直径和连通性的参数化复杂性进行了研究。在积极的一面,我们表明,知道一个大小为$k$的顶点覆盖允许邻接表(AL)流模型中的算法,其传递次数是恒定的,对于任何固定的$k$,内存为$O(\log n)$。这些算法的基础是在$O(k)$通道和$O(k \log n)$内存位中执行宽度优先搜索的方法。在消极的一端,我们展示了许多其他参数导致AL模型中的下界,其中对于任何$p$ -pass算法,即使对于恒定的参数值,也需要$\Omega(n/p)$位的内存。特别是,对于大多数$H$,这适用于具有恒定大小的已知调制器(删除集)的图,该图没有与固定图同构的诱导子图$H$。对于某些情况,我们还可以显示一次通过$\Omega(n \log n)$位的内存下界。我们还证明了二部图上直径的一个更强的$\Omega(n^2/p)$下界。最后,利用我们对流参数化图探索算法的见解,我们展示了一种新的流核化算法,用于计算大小为$k$的顶点覆盖。这将产生一个包含$2k$个顶点(带有$O(k^2)$条边)的内核,在$\text{poly}(k)$通道中作为流生成,并且仅占用$O(k \log n)$位内存。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:604180095
Book学术官方微信