{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T06:26:43Z","timestamp":1773815203138,"version":"3.50.1"},"reference-count":9,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2013,8,21]],"date-time":"2013-08-21T00:00:00Z","timestamp":1377043200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>In this paper, the notion of stability is extended to network flows over time. As a useful device in our proofs, we present an elegant preflow-push variant of the Gale-Shapley algorithm that operates directly on the given network and computes stable flows in pseudo-polynomial time, both in the static flow and the flow over time case. We show periodical properties of stable flows over time on networks with an infinite time horizon. Finally, we discuss the influence of storage at vertices, with different results depending on the priority of the corresponding holdover edges.<\/jats:p>","DOI":"10.3390\/a6030532","type":"journal-article","created":{"date-parts":[[2013,8,21]],"date-time":"2013-08-21T12:49:13Z","timestamp":1377089353000},"page":"532-545","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Stable Flows over Time"],"prefix":"10.3390","volume":"6","author":[{"given":"\u00c1gnes","family":"Cseh","sequence":"first","affiliation":[{"name":"TU Berlin, Institut f\u00a8ur Mathematik, Stra\u00dfe des 17. Juni 136, Berlin 10623, Germany"}]},{"given":"Jannik","family":"Matuschke","sequence":"additional","affiliation":[{"name":"TU Berlin, Institut f\u00a8ur Mathematik, Stra\u00dfe des 17. Juni 136, Berlin 10623, Germany"}]},{"given":"Martin","family":"Skutella","sequence":"additional","affiliation":[{"name":"TU Berlin, Institut f\u00a8ur Mathematik, Stra\u00dfe des 17. Juni 136, Berlin 10623, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2013,8,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1080\/00029890.1962.11989827","article-title":"College admissions and the stability of marriage","volume":"1","author":"Gale","year":"1962","journal-title":"Am. Math. Mon."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1287\/moor.27.3.485.310","article-title":"The stable allocation (or ordinal transportation) problem","volume":"27","author":"Balinski","year":"2002","journal-title":"Math. Oper. Res."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Dean, B.C., Goemans, M.X., and Immorlica, N. (2006). Finite termination of \u201caugmenting path\" algorithms in the presence of irrational problem data.. Algorithms - ESA, 268\u2013279.","DOI":"10.1007\/11841036_26"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1007\/s00453-010-9416-y","article-title":"Faster algorithms for stable allocation problems","volume":"58","author":"Dean","year":"2010","journal-title":"Algorithmica"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"897","DOI":"10.1257\/aer.98.3.897","article-title":"Stability in supply chain networks","volume":"98","author":"Ostrovsky","year":"2008","journal-title":"Am. Econ. 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