Reachability in choice networks

Pub Date : 2023-05-01 DOI:10.1016/j.disopt.2023.100761
Piotr Wojciechowski , K. Subramani , Alvaro Velasquez
{"title":"Reachability in choice networks","authors":"Piotr Wojciechowski ,&nbsp;K. Subramani ,&nbsp;Alvaro Velasquez","doi":"10.1016/j.disopt.2023.100761","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we investigate the problem of determining <span><math><mrow><mi>s</mi><mo>−</mo><mi>t</mi></mrow></math></span> reachability in <strong>choice networks</strong>. In the traditional <span><math><mrow><mi>s</mi><mo>−</mo><mi>t</mi></mrow></math></span> reachability problem, we are given a weighted network tuple <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>〈</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>〉</mo></mrow></mrow></math></span>, with the goal of checking if there exists a path from <span><math><mi>s</mi></math></span> to <span><math><mi>t</mi></math></span> in <span><math><mi>G</mi></math></span>. In an optional choice network, we are given a choice set <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>E</mi><mo>×</mo><mi>E</mi></mrow></math></span>, in addition to the network tuple <span><math><mi>G</mi></math></span>. In the <span><math><mrow><mi>s</mi><mo>−</mo><mi>t</mi></mrow></math></span> reachability problem in choice networks (OCR<span><math><msub><mrow></mrow><mrow><mi>D</mi></mrow></msub></math></span>), the goal is to find whether there exists a path from vertex <span><math><mi>s</mi></math></span> to vertex <span><math><mi>t</mi></math></span>, with the caveat that at most one edge from each edge-pair <span><math><mrow><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>∈</mo><mi>S</mi></mrow></math></span> is used in the path. OCR<span><math><msub><mrow></mrow><mrow><mi>D</mi></mrow></msub></math></span> finds applications in a number of domains, including <strong>routing in wireless networks</strong> and <strong>sensor placement</strong>. We analyze the computational complexities of the OCR<span><math><msub><mrow></mrow><mrow><mi>D</mi></mrow></msub></math></span> problem and its variants from a number of algorithmic perspectives. We show that the problem is <strong>NP-complete</strong> in directed acyclic graphs with bounded pathwidth. Additionally, we show that its optimization version is <strong>NPO PB-complete</strong>. Additionally, we show that the problem is fixed-parameter tractable in the cardinality of the choice set <span><math><mi>S</mi></math></span>. In particular, we show that the problem can be solved in time <span><math><mrow><msup><mrow><mi>O</mi></mrow><mrow><mo>∗</mo></mrow></msup><mrow><mo>(</mo><mn>1</mn><mo>.</mo><mn>4</mn><msup><mrow><mn>2</mn></mrow><mrow><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow></mrow></msup><mo>)</mo></mrow></mrow></math></span>. We also consider weighted versions of the OCR<span><math><msub><mrow></mrow><mrow><mi>D</mi></mrow></msub></math></span> problem and detail their computational complexities; in particular, the optimization version of the <span><math><mrow><mi>W</mi><mi>O</mi><mi>C</mi><msub><mrow><mi>R</mi></mrow><mrow><mi>D</mi></mrow></msub></mrow></math></span> problem is <strong>NPO-complete</strong>. While similar results have been obtained for related problems, our results improve on those results by providing stronger results or by providing results for more limited graph types.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1572528623000038","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

In this paper, we investigate the problem of determining st reachability in choice networks. In the traditional st reachability problem, we are given a weighted network tuple G=V,E,c,s,t, with the goal of checking if there exists a path from s to t in G. In an optional choice network, we are given a choice set SE×E, in addition to the network tuple G. In the st reachability problem in choice networks (OCRD), the goal is to find whether there exists a path from vertex s to vertex t, with the caveat that at most one edge from each edge-pair (x,y)S is used in the path. OCRD finds applications in a number of domains, including routing in wireless networks and sensor placement. We analyze the computational complexities of the OCRD problem and its variants from a number of algorithmic perspectives. We show that the problem is NP-complete in directed acyclic graphs with bounded pathwidth. Additionally, we show that its optimization version is NPO PB-complete. Additionally, we show that the problem is fixed-parameter tractable in the cardinality of the choice set S. In particular, we show that the problem can be solved in time O(1.42|S|). We also consider weighted versions of the OCRD problem and detail their computational complexities; in particular, the optimization version of the WOCRD problem is NPO-complete. While similar results have been obtained for related problems, our results improve on those results by providing stronger results or by providing results for more limited graph types.

分享
查看原文
选择网络的可达性
在本文中,我们研究了在选择网络中确定s−t可达性的问题。在传统的s−t可达性问题中,我们给出了一个加权网络元组G=〈V,E,c,s,t〉,目的是检查G中是否存在从s到t的路径。在可选选择网络中,除了网络元组G之外,我们还得到了一个选择集s⊆E×E,目标是找出是否存在从顶点s到顶点t的路径,但需要注意的是,该路径中最多使用每个边对(x,y)∈s中的一条边。OCRD在许多领域都有应用,包括无线网络中的路由和传感器放置。我们从多个算法角度分析了OCRD问题及其变体的计算复杂性。我们证明了具有有界路径宽度的有向无环图中的问题是NP完全的。此外,我们还展示了它的优化版本是NPO-PB完整的。此外,我们证明了该问题在选择集S的基数中是可处理的固定参数。特别地,我们证明该问题可以在时间O*(1.42|S|)内求解。我们还考虑了OCRD问题的加权版本,并详细说明了它们的计算复杂性;特别地,WOCRD问题的优化版本是NPO完全的。虽然相关问题也得到了类似的结果,但我们的结果通过提供更强的结果或提供更有限的图类型的结果来改进这些结果。
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
×
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:481959085
Book学术官方微信