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Both can be formulated as a submodular function minimization problem.<\/jats:p>\n          <jats:p>We observe that minimizing this submodular function corresponds to a polymatroid sum problem, and using this viewpoint, we give a fast and simple push-relabel algorithm for finding the required sets. This improves on the previously best running time of Murota, Shioura and Yang (ISAAC 2013). Our algorithm is an adaptation of the push-relabel framework by Frank and Mikl\u00f3s (JJIAM 2012) to the particular setting. We obtain a further improvement for the special case of unit-supplies.<\/jats:p>\n          <jats:p>\n            We further show the following monotonicity properties of Walrasian prices: both the minimal and maximal Walrasian prices can only increase if supply of goods decreases, or if the demand of buyers increases. This is derived from a fine-grained analysis of market prices. We call\n            <jats:italic toggle=\"yes\">packing prices<\/jats:italic>\n            a price vector such that there is a feasible allocation where each buyer obtains a utility maximizing set. Conversely, by\n            <jats:italic toggle=\"yes\">covering prices<\/jats:italic>\n            we mean a price vector such that there exists a collection of utility maximizing sets of the buyers that include all available goods. We show that for strong gross substitutes valuations, the component-wise minimal packing prices coincide with the minimal Walrasian prices and the component-wise maximal covering prices coincide with the maximal Walrasian prices. These properties in turn lead to the price monotonicity results.\n          <\/jats:p>","DOI":"10.1145\/3729429","type":"journal-article","created":{"date-parts":[[2025,4,12]],"date-time":"2025-04-12T09:53:21Z","timestamp":1744451601000},"page":"1-47","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":["Faster Dynamic Auctions via Polymatroid Sum"],"prefix":"10.1145","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9809-8821","authenticated-orcid":false,"given":"Katharina","family":"Eickhoff","sequence":"first","affiliation":[{"name":"School of Business and Economics, RWTH Aachen University","place":["Aachen, Germany"]}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3664-3687","authenticated-orcid":false,"given":"Meike","family":"Neuwohner","sequence":"additional","affiliation":[{"name":"Department of Mathematics, London School of Economics and Political Science","place":["London, United Kingdom"]}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8938-8843","authenticated-orcid":false,"given":"Britta","family":"Peis","sequence":"additional","affiliation":[{"name":"School of Business and Economics, RWTH Aachen University","place":["Aachen, Germany"]}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7018-8692","authenticated-orcid":false,"given":"Niklas","family":"Rieken","sequence":"additional","affiliation":[{"name":"School of Business and Economics, RWTH Aachen University","place":["Aachen, Germany"]}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7499-5958","authenticated-orcid":false,"given":"Laura","family":"Vargas Koch","sequence":"additional","affiliation":[{"name":"Research Institute for Discrete Mathematics and Hausdorff Center for Mathematics, University of Bonn","place":["Bonn, Germany"]},{"name":"School of Business and Economics, RWTH Aachen University","place":["Aachen, Germany"]}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1152-200X","authenticated-orcid":false,"given":"L\u00e1szl\u00f3 A.","family":"V\u00e9gh","sequence":"additional","affiliation":[{"name":"London School of Economics and Political Science","place":["London, United Kingdom"]},{"name":"Institute for Advanced Studies, Corvinus University of Budapest","place":["Budapest, Hungary"]},{"name":"Hertz Chair for Algorithms and Optimization, University of Bonn","place":["Bonn, Germany"]}]}],"member":"320","published-online":{"date-parts":[[2025,6,16]]},"reference":[{"doi-asserted-by":"publisher","key":"e_1_3_3_2_2","DOI":"10.21236\/ADA594171"},{"doi-asserted-by":"publisher","unstructured":"Tommy Andersson Christer Andersson and A. 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