{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T12:57:13Z","timestamp":1773406633104,"version":"3.50.1"},"reference-count":16,"publisher":"Wiley","issue":"7","license":[{"start":{"date-parts":[[2006,10,11]],"date-time":"2006-10-11T00:00:00Z","timestamp":1160524800000},"content-version":"vor","delay-in-days":5062,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Networks"],"published-print":{"date-parts":[[1992,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Consider a network <jats:italic>G<\/jats:italic> = <jats:italic>(V,E)<\/jats:italic> with distinguished vertices <jats:italic>s<\/jats:italic> and <jats:italic>t<\/jats:italic>, and with <jats:italic>k<\/jats:italic> different costs on every edge. We consider the problem of finding <jats:italic>k<\/jats:italic> disjoint paths from <jats:italic>s<\/jats:italic> to <jats:italic>t<\/jats:italic> such that the total cost of the paths is minimized, where the <jats:italic>j<\/jats:italic>th edge\u2010cost is associated with the <jats:italic>j<\/jats:italic>th path. The problem has several variants: The paths may be vertex\u2010disjoint or arc\u2010disjoint and the network may be directed or undirected. We show that all four versions of the problem are strongly NP\u2010complete even for <jats:italic>k<\/jats:italic> = 2. We describe polynomial time heuristics for the problem and a polynomial time algorithm for the acyclic directed case.<\/jats:p>","DOI":"10.1002\/net.3230220705","type":"journal-article","created":{"date-parts":[[2007,5,12]],"date-time":"2007-05-12T13:21:11Z","timestamp":1178976071000},"page":"653-667","source":"Crossref","is-referenced-by-count":63,"title":["Finding disjoint paths with different path\u2010costs: Complexity and algorithms"],"prefix":"10.1002","volume":"22","author":[{"given":"Chung\u2010Lun","family":"Li","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"S.","family":"Thomas McCormick","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"David","family":"Simchi\u2010Levi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,10,11]]},"reference":[{"key":"e_1_2_1_2_2","volume-title":"Handbooks in Operations Research and Management Science, Vol. 1, Optimization","author":"Ahuja R. K.","year":"1989"},{"key":"e_1_2_1_3_2","unstructured":"D.BienstockandM. A.Langston Algorithmic implications of the graph minor Theorem. To appear."},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1137\/0205048"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(80)90009-2"},{"key":"e_1_2_1_6_2","volume-title":"Computers and Intractability: A Guide to the Theory of NP\u2010Completeness","author":"Garey M. R.","year":"1979"},{"key":"e_1_2_1_7_2","volume-title":"Graph Theory","author":"Harary F.","year":"1972"},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.1002\/net.3230120306"},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218X(90)90024-7"},{"key":"e_1_2_1_10_2","first-page":"264","article-title":"Constructing disjoint paths on expander graphs","volume":"19","author":"Peleg D.","year":"1987","journal-title":"Symp. Theory Comp."},{"key":"e_1_2_1_11_2","doi-asserted-by":"publisher","DOI":"10.1145\/322047.322048"},{"key":"e_1_2_1_12_2","unstructured":"N.RobertsonandP. D.Seymour Graph minors. XIII. The disjoint paths problem. To appear."},{"key":"e_1_2_1_13_2","doi-asserted-by":"publisher","DOI":"10.1002\/net.3230140405"},{"key":"e_1_2_1_14_2","doi-asserted-by":"publisher","DOI":"10.1145\/322203.322207"},{"key":"e_1_2_1_15_2","doi-asserted-by":"publisher","DOI":"10.1002\/net.3230040204"},{"key":"e_1_2_1_16_2","doi-asserted-by":"publisher","DOI":"10.1002\/net.3230140209"},{"key":"e_1_2_1_17_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0195-6698(80)80039-4"}],"container-title":["Networks"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnet.3230220705","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/net.3230220705","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,23]],"date-time":"2023-10-23T09:40:20Z","timestamp":1698054020000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/net.3230220705"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1992,12]]},"references-count":16,"journal-issue":{"issue":"7","published-print":{"date-parts":[[1992,12]]}},"alternative-id":["10.1002\/net.3230220705"],"URL":"https:\/\/doi.org\/10.1002\/net.3230220705","archive":["Portico"],"relation":{},"ISSN":["0028-3045","1097-0037"],"issn-type":[{"value":"0028-3045","type":"print"},{"value":"1097-0037","type":"electronic"}],"subject":[],"published":{"date-parts":[[1992,12]]}}}