{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,22]],"date-time":"2025-10-22T23:38:07Z","timestamp":1761176287108,"version":"build-2065373602"},"reference-count":0,"publisher":"IOS Press","isbn-type":[{"value":"9781643686318","type":"electronic"}],"license":[{"start":{"date-parts":[[2025,10,21]],"date-time":"2025-10-21T00:00:00Z","timestamp":1761004800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,10,21]]},"abstract":"<jats:p>Many optimization algorithms represent solutions as permutations. However, despite their apparent simplicity, permutations pose significant challenges\u2014especially for Global Random Search (GRS) algorithms\u2014due to the mutual-exclusivity constraint. This constraint complicates both the learning and sampling of probability distributions over the permutation space, often leading to computationally expensive procedures. A promising alternative involves transforming permutation-encoded solutions into integer vectors using bijective functions on the symmetric group Sn, resulting in what are known as inversion vectors. While inversion vectors have been studied for centuries, a unified and formal framework encompassing all their codifications has been lacking. In this paper, we introduce precise definitions and a unified notation for various types of inversion vector codifications. We establish bijective transformations between them, providing a formal characterization of their relationships and properties. Leveraging this theoretical foundation, we analyze and explain the behavior of GRS algorithms across different permutation problems when using different inversion vector representations.<\/jats:p>","DOI":"10.3233\/faia251382","type":"book-chapter","created":{"date-parts":[[2025,10,22]],"date-time":"2025-10-22T09:59:59Z","timestamp":1761127199000},"source":"Crossref","is-referenced-by-count":0,"title":["A Unified View of Bijective Transformations for Optimizing Permutation Problems"],"prefix":"10.3233","author":[{"given":"Mikel","family":"Malagon","sequence":"first","affiliation":[{"name":"University of the Basque Country UPV\/EHU"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Aimar","family":"Barrena","sequence":"additional","affiliation":[{"name":"University of the Basque Country UPV\/EHU"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hugo","family":"I\u00f1igo","sequence":"additional","affiliation":[{"name":"University of the Basque Country UPV\/EHU"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ekhi\u00f1e","family":"Irurozki","sequence":"additional","affiliation":[{"name":"Telecom Paris"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jose A.","family":"Lozano","sequence":"additional","affiliation":[{"name":"University of the Basque Country UPV\/EHU"},{"name":"Basque Center for Applied Mathematics (BCAM)"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Josu","family":"Ceberio","sequence":"additional","affiliation":[{"name":"University of the Basque Country UPV\/EHU"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"7437","container-title":["Frontiers in Artificial Intelligence and Applications","ECAI 2025"],"original-title":[],"link":[{"URL":"https:\/\/ebooks.iospress.nl\/pdf\/doi\/10.3233\/FAIA251382","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,22]],"date-time":"2025-10-22T09:59:59Z","timestamp":1761127199000},"score":1,"resource":{"primary":{"URL":"https:\/\/ebooks.iospress.nl\/doi\/10.3233\/FAIA251382"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,21]]},"ISBN":["9781643686318"],"references-count":0,"URL":"https:\/\/doi.org\/10.3233\/faia251382","relation":{},"ISSN":["0922-6389","1879-8314"],"issn-type":[{"value":"0922-6389","type":"print"},{"value":"1879-8314","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,21]]}}}