{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:23Z","timestamp":1753893803290,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>For a given shape $S$ in the plane, one can ask what is the lowest possible density of a point set $P$ that pierces (\"intersects\", \"hits\") all translates of $S$. This is equivalent to determining the covering density of $S$ and as such is well studied. Here we study the analogous question for families of shapes where the connection to covering is altered. That is, we require that a single point set $P$ simultaneously pierces each translate of each shape from some family $\\mathcal{F}$. We denote the lowest possible density of such an $\\mathcal{F}$-piercing point set by $\\pi_T(\\mathcal{F})$. Specifically, we focus on families $\\mathcal{F}$ consisting of axis-parallel rectangles. When $|\\mathcal{F}|=2$ we exactly solve the case when one rectangle is more squarish than $2\\times 1$, and give bounds (within $10\\,\\%$ of each other) for the remaining case when one rectangle is wide and the other one is tall. When $|\\mathcal{F}|\\ge 2$ we present a linear-time constant-factor approximation algorithm for computing $\\pi_T(\\mathcal{F})$ (with ratio $1.895$).<\/jats:p>","DOI":"10.37236\/12041","type":"journal-article","created":{"date-parts":[[2024,2,9]],"date-time":"2024-02-09T15:45:31Z","timestamp":1707493531000},"source":"Crossref","is-referenced-by-count":0,"title":["Piercing All Translates of a Set of Axis-Parallel Rectangles"],"prefix":"10.37236","volume":"31","author":[{"given":"Adrian","family":"Dumitrescu","sequence":"first","affiliation":[]},{"given":"Josef","family":"Tkadlec","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2024,2,9]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i1p33\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i1p33\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,9]],"date-time":"2024-02-09T15:45:31Z","timestamp":1707493531000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v31i1p33"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,9]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,1,12]]}},"URL":"https:\/\/doi.org\/10.37236\/12041","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2024,2,9]]},"article-number":"P1.33"}}