Piotr Wojciechowski , K. Subramani , Alvaro Velasquez
{"title":"Reachability in choice networks","authors":"Piotr Wojciechowski , K. Subramani , 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":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Optimization","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1572528623000038","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we investigate the problem of determining reachability in choice networks. In the traditional reachability problem, we are given a weighted network tuple , with the goal of checking if there exists a path from to in . In an optional choice network, we are given a choice set , in addition to the network tuple . In the reachability problem in choice networks (OCR), the goal is to find whether there exists a path from vertex to vertex , with the caveat that at most one edge from each edge-pair is used in the path. OCR finds applications in a number of domains, including routing in wireless networks and sensor placement. We analyze the computational complexities of the OCR 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 . In particular, we show that the problem can be solved in time . We also consider weighted versions of the OCR problem and detail their computational complexities; in particular, the optimization version of the 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.
期刊介绍:
Discrete Optimization publishes research papers on the mathematical, computational and applied aspects of all areas of integer programming and combinatorial optimization. In addition to reports on mathematical results pertinent to discrete optimization, the journal welcomes submissions on algorithmic developments, computational experiments, and novel applications (in particular, large-scale and real-time applications). The journal also publishes clearly labelled surveys, reviews, short notes, and open problems. Manuscripts submitted for possible publication to Discrete Optimization should report on original research, should not have been previously published, and should not be under consideration for publication by any other journal.