In the equal maximum flow problem (EMFP), we aim for a maximum flow where we require the same flow value on all arcs in some given subsets of the arc set, so called homologous arc sets. In this article, we study the closely related almost equal maximum flow problems (AEMFP) where the flow values on arcs of one homologous arc set differ at most by the valuation of a so called deviation function <mjx-container aria-label="normal upper Delta" ctxtmenu_counter="0" ctxtmenu_oldtabindex="1" jax="CHTML" role="application" sre-explorer- style="font-size: 103%; position: relative;" tabindex="0"><mjx-math aria-hidden="true" location="graphic/net22209-math-0001.png"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation="clearspeak:simple" data-semantic-font="normal" data-semantic- data-semantic-role="greekletter" data-semantic-speech="normal upper Delta" data-semantic-type="identifier"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden="true" display="inline" unselectable="on"><math altimg="urn:x-wiley:net:media:net22209:net22209-math-0001" display="inline" location="graphic/net22209-math-0001.png" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi data-semantic-="" data-semantic-annotation="clearspeak:simple" data-semantic-font="normal" data-semantic-role="greekletter" data-semantic-speech="normal upper Delta" data-semantic-type="identifier" mathvariant="normal">Δ</mi></mrow>$$ Delta $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>. We prove that the integer AEMFP is in general <mjx-container aria-label="script upper N script upper P" ctxtmenu_counter="1" ctxtmenu_oldtabindex="1" jax="CHTML" role="application" sre-explorer- style="font-size: 103%; position: relative;" tabindex="0"><mjx-math aria-hidden="true" location="graphic/net22209-math-0002.png"><mjx-mrow data-semantic-annotation="clearspeak:simple;clearspeak:unit" data-semantic-children="0,1" data-semantic-content="2" data-semantic- data-semantic-role="implicit" data-semantic-speech="script upper N script upper P" data-semantic-type="infixop"><mjx-mi data-semantic-annotation="clearspeak:simple" data-semantic-font="script" data-semantic- data-semantic-parent="3" data-semantic-role="latinletter" data-semantic-type="identifier"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added="true" data-semantic- data-semantic-operator="infixop," data-semantic-parent="3" data-semantic-role="multiplication" data-semantic-type="operator" style="margin-left: 0.056em; margin-right: 0.056em;"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation="clearspeak:simple" data-semantic-font="script" data-semantic- data-semantic-parent="3" data-semantic-role="latinletter" data-semantic-type="identifier"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-math><mjx-assistive-mml aria-hidden="true" display="inline" unselectable="on"><math altimg="urn:x-wiley:net:media:net22209:net22209-math-0002" display="inline" location="graphic/net22209-
{"title":"Algorithms and complexity for the almost equal maximum flow problem","authors":"Rebekka Haese, Till Heller, Sven O. Krumke","doi":"10.1002/net.22209","DOIUrl":"https://doi.org/10.1002/net.22209","url":null,"abstract":"In the equal maximum flow problem (EMFP), we aim for a maximum flow where we require the same flow value on all arcs in some given subsets of the arc set, so called homologous arc sets. In this article, we study the closely related almost equal maximum flow problems (AEMFP) where the flow values on arcs of one homologous arc set differ at most by the valuation of a so called deviation function <mjx-container aria-label=\"normal upper Delta\" ctxtmenu_counter=\"0\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/net22209-math-0001.png\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-role=\"greekletter\" data-semantic-speech=\"normal upper Delta\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:net:media:net22209:net22209-math-0001\" display=\"inline\" location=\"graphic/net22209-math-0001.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-role=\"greekletter\" data-semantic-speech=\"normal upper Delta\" data-semantic-type=\"identifier\" mathvariant=\"normal\">Δ</mi></mrow>$$ Delta $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>. We prove that the integer AEMFP is in general <mjx-container aria-label=\"script upper N script upper P\" ctxtmenu_counter=\"1\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/net22209-math-0002.png\"><mjx-mrow data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"0,1\" data-semantic-content=\"2\" data-semantic- data-semantic-role=\"implicit\" data-semantic-speech=\"script upper N script upper P\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"script\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"3\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"script\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:net:media:net22209:net22209-math-0002\" display=\"inline\" location=\"graphic/net22209-","PeriodicalId":54734,"journal":{"name":"Networks","volume":"205 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139554307","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Marta Baldomero-Naranjo, Jörg Kalcsics, Antonio M. Rodríguez-Chía
In this article, we study the complexity of the upgrading version of the maximal covering location problem with edge length modifications on networks. This problem is NP-hard on general networks. However, in some particular cases, we prove that this problem is solvable in polynomial time. The cases of star and path networks combined with different assumptions for the model parameters are analysed. In particular, we obtain that the problem on star networks is solvable in