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Program."],"published-print":{"date-parts":[[2024,3]]},"abstract":"Abstract<\/jats:title>We investigate the parameterized complexity of finding diverse sets of solutions to three fundamental combinatorial problems. The input to the Weighted Diverse Bases<\/jats:sc> problem consists of a matroid $$M$$<\/jats:tex-math>\n M<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, a weight function $$\\omega :E(M)\\rightarrow \\mathbb {N} $$<\/jats:tex-math>\n \n \u03c9<\/mml:mi>\n :<\/mml:mo>\n E<\/mml:mi>\n (<\/mml:mo>\n M<\/mml:mi>\n )<\/mml:mo>\n \u2192<\/mml:mo>\n N<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and integers $$k\\ge 1, d\\ge 1$$<\/jats:tex-math>\n \n k<\/mml:mi>\n \u2265<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n d<\/mml:mi>\n \u2265<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The task is to decide if there is a collection of $$k$$<\/jats:tex-math>\n k<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>bases<\/jats:italic>$$B_{1}, \\dotsc , B_{k}$$<\/jats:tex-math>\n \n \n B<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \u22ef<\/mml:mo>\n ,<\/mml:mo>\n \n B<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of $$M$$<\/jats:tex-math>\n M<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> such that the weight of the symmetric difference of any pair of these bases is at least $$d$$<\/jats:tex-math>\n d<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The input to the Weighted Diverse Common Independent Sets<\/jats:sc> problem consists of two matroids $$M_{1},M_{2}$$<\/jats:tex-math>\n \n \n M<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n M<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> defined on the same ground set $$E$$<\/jats:tex-math>\n E<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, a weight function $$\\omega :E\\rightarrow \\mathbb {N} $$<\/jats:tex-math>\n \n \u03c9<\/mml:mi>\n :<\/mml:mo>\n E<\/mml:mi>\n \u2192<\/mml:mo>\n N<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and integers $$k\\ge 1, d\\ge 1$$<\/jats:tex-math>\n \n k<\/mml:mi>\n \u2265<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n d<\/mml:mi>\n \u2265<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The task is to decide if there is a collection of $$k$$<\/jats:tex-math>\n k<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>common independent sets<\/jats:italic>$$I_{1}, \\dotsc , I_{k}$$<\/jats:tex-math>\n \n \n I<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \u22ef<\/mml:mo>\n ,<\/mml:mo>\n \n I<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of $$M_{1}$$<\/jats:tex-math>\n \n M<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$M_{2}$$<\/jats:tex-math>\n \n M<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> such that the weight of the symmetric difference of any pair of these sets is at least $$d$$<\/jats:tex-math>\n d<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The input to the Diverse Perfect Matchings<\/jats:sc> problem consists of a graph $$G$$<\/jats:tex-math>\n G<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and integers $$k\\ge 1, d\\ge 1$$<\/jats:tex-math>\n \n k<\/mml:mi>\n \u2265<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n d<\/mml:mi>\n \u2265<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The task is to decide if $$G$$<\/jats:tex-math>\n G<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> contains $$k$$<\/jats:tex-math>\n k<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>perfect matchings<\/jats:italic>$$M_{1},\\dotsc ,M_{k}$$<\/jats:tex-math>\n \n \n M<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \u22ef<\/mml:mo>\n ,<\/mml:mo>\n \n M<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> such that the symmetric difference of any two of these matchings is at least $$d$$<\/jats:tex-math>\n d<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We show that none of these problems can be solved in polynomial time unless $${{\\,\\mathrm{\\textsf{P}}\\,}} ={{\\,\\mathrm{\\textsf{NP}}\\,}} $$<\/jats:tex-math>\n \n \n \n P<\/mml:mi>\n \n <\/mml:mrow>\n =<\/mml:mo>\n \n \n NP<\/mml:mi>\n \n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We derive fixed-parameter tractable ($${{\\,\\mathrm{\\textsf{FPT}}\\,}}$$<\/jats:tex-math>\n \n \n FPT<\/mml:mi>\n \n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>) algorithms for all three problems with $$(k,d)$$<\/jats:tex-math>\n \n (<\/mml:mo>\n k<\/mml:mi>\n ,<\/mml:mo>\n d<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> as the parameter, and present a $$poly(k,d)$$<\/jats:tex-math>\n \n p<\/mml:mi>\n o<\/mml:mi>\n l<\/mml:mi>\n y<\/mml:mi>\n (<\/mml:mo>\n k<\/mml:mi>\n ,<\/mml:mo>\n d<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-sized kernel for Weighted Diverse Bases<\/jats:sc>.\n<\/jats:p>","DOI":"10.1007\/s10107-023-01959-z","type":"journal-article","created":{"date-parts":[[2023,4,15]],"date-time":"2023-04-15T11:02:02Z","timestamp":1681556522000},"page":"415-447","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Diverse collections in matroids and graphs"],"prefix":"10.1007","volume":"204","author":[{"given":"Fedor V.","family":"Fomin","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2619-2990","authenticated-orcid":false,"given":"Petr A.","family":"Golovach","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fahad","family":"Panolan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Geevarghese","family":"Philip","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Saket","family":"Saurabh","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,4,15]]},"reference":[{"key":"1959_CR1","doi-asserted-by":"publisher","DOI":"10.1016\/j.artint.2021.103644","volume":"303","author":"J Baste","year":"2022","unstructured":"Baste, J., Fellows, M.R., Jaffke, L., Masar\u00edk, T., de Oliveira Oliveira, M., Philip, G., Rosamond, F.A.: Diversity of solutions: an exploration through the lens of fixed-parameter tractability theory. 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