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The task is to compute a new hypergraph, called a kernel<\/jats:italic>, whose size is polynomial with respect to the parameter\u00a0k<\/jats:italic> and which has a size-k<\/jats:italic> hitting set if, and only if, the original hypergraph has one. State-of-the-art algorithms compute kernels of size\u00a0$$k^d$$<\/jats:tex-math>\n \n k<\/mml:mi>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> (which is a polynomial as d<\/jats:italic> is a constant), and they do so in time $$m\\cdot 2^d {\\text {poly}}(d)$$<\/jats:tex-math>\n \n m<\/mml:mi>\n \u00b7<\/mml:mo>\n \n 2<\/mml:mn>\n d<\/mml:mi>\n <\/mml:msup>\n poly<\/mml:mtext>\n \n (<\/mml:mo>\n d<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for a small polynomial $${\\text {poly}}(d)$$<\/jats:tex-math>\n \n poly<\/mml:mtext>\n (<\/mml:mo>\n d<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> (which is linear in the hypergraph size for d<\/jats:italic> fixed). We generalize this task to the dynamic<\/jats:italic> setting where hyperedges may continuously be added or deleted and one constantly has to keep track of a size-$$k^d$$<\/jats:tex-math>\n \n k<\/mml:mi>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> kernel. This paper presents a deterministic<\/jats:italic> solution with worst-case<\/jats:italic> time $$3^d {\\text {poly}}(d)$$<\/jats:tex-math>\n \n \n 3<\/mml:mn>\n d<\/mml:mi>\n <\/mml:msup>\n poly<\/mml:mtext>\n \n (<\/mml:mo>\n d<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for updating the kernel upon inserts and time\u00a0$$5^d {\\text {poly}}(d)$$<\/jats:tex-math>\n \n \n 5<\/mml:mn>\n d<\/mml:mi>\n <\/mml:msup>\n poly<\/mml:mtext>\n \n (<\/mml:mo>\n d<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for updates upon deletions. These bounds nearly match the time $$2^d {\\text {poly}}(d)$$<\/jats:tex-math>\n \n \n 2<\/mml:mn>\n d<\/mml:mi>\n <\/mml:msup>\n poly<\/mml:mtext>\n \n (<\/mml:mo>\n d<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> needed by the best static algorithm per hyperedge. Let us stress that for constant\u00a0d<\/jats:italic> our algorithm maintains a hitting set kernel with constant, deterministic, worst-case<\/jats:italic> update time that is independent of n<\/jats:italic>, m<\/jats:italic>, and the parameter\u00a0k<\/jats:italic>. As a consequence, we also get a deterministic dynamic algorithm for keeping track of size-k<\/jats:italic> hitting sets in d<\/jats:italic>-hypergraphs with update times O<\/jats:italic>(1) and query times $$O(c^k)$$<\/jats:tex-math>\n \n O<\/mml:mi>\n (<\/mml:mo>\n \n c<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> where $$c = d - 1 + O(1\/d)$$<\/jats:tex-math>\n \n c<\/mml:mi>\n =<\/mml:mo>\n d<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n +<\/mml:mo>\n O<\/mml:mi>\n (<\/mml:mo>\n 1<\/mml:mn>\n \/<\/mml:mo>\n d<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> equals the best base known for the static setting.<\/jats:p>","DOI":"10.1007\/s00453-022-00986-0","type":"journal-article","created":{"date-parts":[[2022,6,22]],"date-time":"2022-06-22T05:03:46Z","timestamp":1655874226000},"page":"3459-3488","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Dynamic Kernels for Hitting Sets and Set Packing"],"prefix":"10.1007","volume":"84","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4566-8031","authenticated-orcid":false,"given":"Max","family":"Bannach","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zacharias","family":"Heinrich","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"R\u00fcdiger","family":"Reischuk","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Till","family":"Tantau","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2022,6,22]]},"reference":[{"issue":"7","key":"986_CR1","doi-asserted-by":"publisher","first-page":"524","DOI":"10.1016\/j.jcss.2009.09.002","volume":"76","author":"FN Abu-Khzam","year":"2010","unstructured":"Abu-Khzam, F.N.: A Kernelization Algorithm for d-Hitting Set. 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