Benjamin Bergougnoux, Oscar Defrain, Fionn Mc Inerney
{"title":"Enumerating Minimal Solution Sets for Metric Graph Problems","authors":"Benjamin Bergougnoux, Oscar Defrain, 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.
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.