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# Online and Dynamic Algorithms for Geometric Set Cover and Hitting Set

Mar 2023

Set cover and hitting set are fundamental problems in combinatorialoptimization which are well-studied in the offline, online, and dynamicsettings. We study the geometric versions of these problems and present newonline and dynamic algorithms for them. In the online version of set cover(resp. hitting set), $m$ sets (resp.~$n$ points) are give $n$ points (resp.~$m$sets) arrive online, one-by-one. In the dynamic versions, points (resp. sets)can arrive as well as depart. Our goal is to maintain a set cover (resp.hitting set), minimizing the size of the computed solution. For online set cover for (axis-parallel) squares of arbitrary sizes, wepresent a tight $O(\log n)$-competitive algorithm. In the same setting forhitting set, we provide a tight $O(\log N)$-competitive algorithm, assumingthat all points have integral coordinates in $[0,N)^{2}$. No online algorithmhad been known for either of these settings, not even for unit squares (apartfrom the known online algorithms for arbitrary set systems). For both dynamic set cover and hitting set with $d$-dimensionalhyperrectangles, we obtain $(\log m)^{O(d)}$-approximation algorithms with$(\log m)^{O(d)}$ worst-case update time. This partially answers an openquestion posed by Chan et al. [SODA'22]. Previously, no dynamic algorithms withpolylogarithmic update time were known even in the setting of squares (foreither of these problems). Our main technical contributions are an\emph{extended quad-tree }approach and a \emph{frequency reduction} techniquethat reduces geometric set cover instances to instances of general set coverwith bounded frequency.

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