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The existing work on Lagrangean algorithms to the -hard problem of finding minimum weight stable spanning trees is limited to relaxations with the integrality property. We exploit a new relaxation of this problem: fixed cardinality stable sets in the underlying conflict graph <jats:inline-formula><jats:alternatives><jats:tex-math>$$H =(E,C)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>H<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>E<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>C<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We find interesting properties of the corresponding polytope, and determine stronger dual bounds in a Lagrangean decomposition framework, optimizing over the spanning tree polytope of <jats:italic>G<\/jats:italic> and the fixed cardinality stable set polytope of <jats:italic>H<\/jats:italic> in the subproblems. This is equivalent to dualizing exponentially many subtour elimination constraints, while limiting the number of multipliers in the dual problem to |<jats:italic>E<\/jats:italic>|. It is also a proof of concept for combining Lagrangean relaxation with the power of integer programming solvers over strongly NP-hard subproblems. We present encouraging computational results using a dual method that comprises the Volume Algorithm, initialized with multipliers determined by Lagrangean dual-ascent. In particular, the bound is within 5.5% of the optimum in 146 out of 200 benchmark instances; it actually matches the optimum in 75 cases. All of the implementation is made available in a free, open-source repository.\n<\/jats:p>","DOI":"10.1007\/s11590-022-01949-8","type":"journal-article","created":{"date-parts":[[2022,11,11]],"date-time":"2022-11-11T03:02:41Z","timestamp":1668135761000},"page":"1317-1335","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Polyhedral results and stronger Lagrangean bounds for stable spanning trees"],"prefix":"10.1007","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9007-0237","authenticated-orcid":false,"given":"Phillippe","family":"Samer","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1110-3382","authenticated-orcid":false,"given":"Dag","family":"Haugland","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,11,11]]},"reference":[{"issue":"3","key":"1949_CR1","doi-asserted-by":"publisher","first-page":"399","DOI":"10.1002\/1520-6750(199204)39:33.0.CO;2-0","volume":"39","author":"A Assad","year":"1992","unstructured":"Assad, A., Xu, W.: The quadratic minimum spanning tree problem. 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