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":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 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.