Enumerating Minimal Solution Sets for Metric Graph Problems

IF 0.9 4区 计算机科学 Q4 COMPUTER SCIENCE, SOFTWARE ENGINEERING
Benjamin Bergougnoux, Oscar Defrain, Fionn Mc Inerney
{"title":"Enumerating Minimal Solution Sets for Metric Graph Problems","authors":"Benjamin Bergougnoux,&nbsp;Oscar Defrain,&nbsp;Fionn Mc Inerney","doi":"10.1007/s00453-025-01300-4","DOIUrl":null,"url":null,"abstract":"<div><p>Problems from metric graph theory like <span>Metric Dimension</span>, <span>Geodetic Set</span>, and <span>Strong Metric Dimension</span> have recently had an impact in parameterized complexity by being the first known problems in <span>NP</span> to admit double-exponential lower bounds in the treewidth, and even in the vertex cover number for the latter, assuming the Exponential Time Hypothesis. We initiate the study of enumerating minimal solution sets for these problems and show that they are also of great interest in enumeration. Specifically, we show that enumerating minimal resolving sets in graphs and minimal geodetic sets in split graphs are equivalent to enumerating minimal transversals in hypergraphs (denoted <span>Trans-Enum</span>), whose solvability in total-polynomial time is one of the most important open problems in algorithmic enumeration. This provides two new natural examples to a question that emerged in recent works: for which vertex (or edge) set graph property <span>\\(\\Pi \\)</span> is the enumeration of minimal (or maximal) subsets satisfying <span>\\(\\Pi \\)</span> equivalent to <span>Trans-Enum</span>? As very few properties are known to fit within this context—namely, those related to minimal domination—our results make significant progress in characterizing such properties, and provide new angles to approach <span>Trans-Enum</span>. In contrast, we observe that minimal strong resolving sets can be enumerated with polynomial delay. Additionally, we consider cases where our reductions do not apply, namely graphs with no long induced paths, and show both positive and negative results related to the enumeration and extension of partial solutions.</p></div>","PeriodicalId":50824,"journal":{"name":"Algorithmica","volume":"87 5","pages":"712 - 735"},"PeriodicalIF":0.9000,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algorithmica","FirstCategoryId":"94","ListUrlMain":"https://link.springer.com/article/10.1007/s00453-025-01300-4","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, SOFTWARE ENGINEERING","Score":null,"Total":0}
引用次数: 0

Abstract

Problems from metric graph theory like Metric Dimension, Geodetic Set, and Strong Metric Dimension have recently had an impact in parameterized complexity by being the first known problems in NP to admit double-exponential lower bounds in the treewidth, and even in the vertex cover number for the latter, assuming the Exponential Time Hypothesis. We initiate the study of enumerating minimal solution sets for these problems and show that they are also of great interest in enumeration. Specifically, we show that enumerating minimal resolving sets in graphs and minimal geodetic sets in split graphs are equivalent to enumerating minimal transversals in hypergraphs (denoted Trans-Enum), whose solvability in total-polynomial time is one of the most important open problems in algorithmic enumeration. This provides two new natural examples to a question that emerged in recent works: for which vertex (or edge) set graph property \(\Pi \) is the enumeration of minimal (or maximal) subsets satisfying \(\Pi \) equivalent to Trans-Enum? As very few properties are known to fit within this context—namely, those related to minimal domination—our results make significant progress in characterizing such properties, and provide new angles to approach Trans-Enum. In contrast, we observe that minimal strong resolving sets can be enumerated with polynomial delay. Additionally, we consider cases where our reductions do not apply, namely graphs with no long induced paths, and show both positive and negative results related to the enumeration and extension of partial solutions.

度量图问题的最小解集枚举
度量图论中的问题,如度量维数、大地测量集和强度量维数,最近对参数化复杂性产生了影响,因为它们是NP中第一个承认树宽双指数下界的问题,甚至是后者的顶点覆盖数,假设指数时间假设。我们开始了对这些问题的枚举最小解集的研究,并表明它们在枚举中也有很大的兴趣。具体地说,我们证明了图中的最小解析集和分割图中的最小测地集的枚举等价于超图中的最小截线的枚举,其在全多项式时间内的可解性是算法枚举中最重要的开放问题之一。这为最近出现的一个问题提供了两个新的自然例子:对于哪个顶点(或边)集合图属性\(\Pi \)是满足\(\Pi \)等价于泛枚举的最小(或最大)子集的枚举?由于很少有已知的属性适合这种情况-即与最小支配相关的属性-我们的结果在表征此类属性方面取得了重大进展,并为研究Trans-Enum提供了新的角度。相反,我们观察到最小强解析集可以用多项式延迟枚举。此外,我们考虑了我们的约简不适用的情况,即没有长诱导路径的图,并显示了与部分解的枚举和扩展相关的正负结果。
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
来源期刊
Algorithmica
Algorithmica 工程技术-计算机:软件工程
CiteScore
2.80
自引率
9.10%
发文量
158
审稿时长
12 months
期刊介绍: Algorithmica is an international journal which publishes theoretical papers on algorithms that address problems arising in practical areas, and experimental papers of general appeal for practical importance or techniques. The development of algorithms is an integral part of computer science. The increasing complexity and scope of computer applications makes the design of efficient algorithms essential. Algorithmica covers algorithms in applied areas such as: VLSI, distributed computing, parallel processing, automated design, robotics, graphics, data base design, software tools, as well as algorithms in fundamental areas such as sorting, searching, data structures, computational geometry, and linear programming. In addition, the journal features two special sections: Application Experience, presenting findings obtained from applications of theoretical results to practical situations, and Problems, offering short papers presenting problems on selected topics of computer science.
×
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