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Program."],"published-print":{"date-parts":[[2025,3]]},"abstract":"Abstract<\/jats:title>\n There has been significant work recently on integer programs (IPs) \n \n $$\\min \\{c^\\top x :Ax\\le b,\\,x\\in \\mathbb {Z}^n\\}$$<\/jats:tex-math>\n \n \n min<\/mml:mo>\n {<\/mml:mo>\n \n c<\/mml:mi>\n \u22a4<\/mml:mi>\n <\/mml:msup>\n x<\/mml:mi>\n :<\/mml:mo>\n A<\/mml:mi>\n x<\/mml:mi>\n \u2264<\/mml:mo>\n b<\/mml:mi>\n ,<\/mml:mo>\n \n x<\/mml:mi>\n \u2208<\/mml:mo>\n \n \n Z<\/mml:mi>\n <\/mml:mrow>\n n<\/mml:mi>\n <\/mml:msup>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> with a constraint marix A<\/jats:italic> with bounded subdeterminants. This is motivated by a well-known conjecture claiming that, for any constant \n \n $$\\Delta \\in \\mathbb {Z}_{>0}$$<\/jats:tex-math>\n \n \n \u0394<\/mml:mi>\n \u2208<\/mml:mo>\n \n Z<\/mml:mi>\n \n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, \n \n $$\\Delta $$<\/jats:tex-math>\n \n \u0394<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-modular IPs are efficiently solvable, which are IPs where the constraint matrix \n \n $$A\\in \\mathbb {Z}^{m\\times n}$$<\/jats:tex-math>\n \n \n A<\/mml:mi>\n \u2208<\/mml:mo>\n \n \n Z<\/mml:mi>\n <\/mml:mrow>\n \n m<\/mml:mi>\n \u00d7<\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> has full column rank and all \n \n $$n\\times n$$<\/jats:tex-math>\n \n \n n<\/mml:mi>\n \u00d7<\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> minors of A<\/jats:italic> are within \n \n $$\\{-\\Delta , \\dots , \\Delta \\}$$<\/jats:tex-math>\n \n \n {<\/mml:mo>\n -<\/mml:mo>\n \u0394<\/mml:mi>\n ,<\/mml:mo>\n \u22ef<\/mml:mo>\n ,<\/mml:mo>\n \u0394<\/mml:mi>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. Previous progress on this question, in particular for \n \n $$\\Delta =2$$<\/jats:tex-math>\n \n \n \u0394<\/mml:mi>\n =<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, relies on algorithms that solve an important special case, namely strictly<\/jats:italic>\n \n \n $$\\Delta $$<\/jats:tex-math>\n \n \u0394<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-modular IPs<\/jats:italic>, which further restrict the \n \n $$n\\times n$$<\/jats:tex-math>\n \n \n n<\/mml:mi>\n \u00d7<\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> minors of A<\/jats:italic> to be within \n \n $$\\{-\\Delta , 0, \\Delta \\}$$<\/jats:tex-math>\n \n \n {<\/mml:mo>\n -<\/mml:mo>\n \u0394<\/mml:mi>\n ,<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n \u0394<\/mml:mi>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. Even for \n \n $$\\Delta =2$$<\/jats:tex-math>\n \n \n \u0394<\/mml:mi>\n =<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, such problems include well-known combinatorial optimization problems like the minimum odd\/even cut problem. The conjecture remains open even for strictly \n \n $$\\Delta $$<\/jats:tex-math>\n \n \u0394<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-modular IPs. Prior advances were restricted to prime \n \n $$\\Delta $$<\/jats:tex-math>\n \n \u0394<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, which allows for employing strong number-theoretic results. In this work, we make first progress beyond the prime case by presenting techniques not relying on such strong number-theoretic prime results. In particular, our approach implies that there is a randomized algorithm to check feasibility of strictly \n \n $$\\Delta $$<\/jats:tex-math>\n \n \u0394<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-modular IPs in strongly polynomial time if \n \n $$\\Delta \\le 4$$<\/jats:tex-math>\n \n \n \u0394<\/mml:mi>\n \u2264<\/mml:mo>\n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s10107-024-02148-2","type":"journal-article","created":{"date-parts":[[2024,10,30]],"date-time":"2024-10-30T16:15:01Z","timestamp":1730304901000},"page":"731-760","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Advances on strictly $$\\Delta $$-modular IPs"],"prefix":"10.1007","volume":"210","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3059-6402","authenticated-orcid":false,"given":"Martin","family":"N\u00e4gele","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6864-1953","authenticated-orcid":false,"given":"Christian","family":"N\u00f6bel","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3515-4953","authenticated-orcid":false,"given":"Richard","family":"Santiago","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7148-9304","authenticated-orcid":false,"given":"Rico","family":"Zenklusen","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,10,30]]},"reference":[{"key":"2148_CR1","doi-asserted-by":"publisher","unstructured":"Artmann, S., Weismantel, R., Zenklusen, R.: A strongly polynomial algorithm for bimodular integer linear programming. 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