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DTSTART;TZID=Asia/Seoul:20200512T163000
DTEND;TZID=Asia/Seoul:20200512T173000
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SUMMARY:Eun Jung Kim (김은정)\, Twin-width: tractable FO model checking
DESCRIPTION:Inspired by a width invariant defined on permutations by Guillemot and Marx [SODA ’14]\, we introduce the notion of twin-width on graphs and on matrices. Proper minor-closed classes\, bounded rank-width graphs\, map graphs\, $K_t$-free unit $d$-dimensional ball graphs\, posets with antichains of bounded size\, and proper subclasses of dimension-2 posets all have bounded twin-width. On all these classes (except map graphs without geometric embedding) we show how to compute in polynomial time a sequence of $d$-contractions\, witness that the twin-width is at most $d$. We show that FO model checking\, that is deciding if a given first-order formula $\phi$ evaluates to true for a given binary structure $G$ on a domain $D$\, is FPT in $|\phi|$ on classes of bounded twin-width\, provided the witness is given. More precisely\, being given a $d$-contraction sequence for $G$\, our algorithm runs in time $f(d\,|\phi|) \cdot |D|$ where $f$ is a computable but non-elementary function. We also prove that bounded twin-width is preserved by FO interpretations and transductions (allowing operations such as squaring or complementing a graph). This unifies and significantly extends the knowledge on fixed-parameter tractability of FO model checking on non-monotone classes\, such as the FPT algorithm on bounded-width posets by Gajarský et al. [FOCS ’15]. \nIn order to explore the limits of twin-width\, we generalize to bounded twin-width classes a result by Norine et al. [JCTB ’06] stating that proper minor-free classes are small (i.e.\, they contain at most $n! c^n$ graphs on $n$ vertices\, for some constant $c$). This implies by a counting argument that bounded-degree graphs\, interval graphs\, and unit disk graphs have unbounded twin-width. \nJoint work with Stéphan Thomassé\, Édouard Bonnet\, and Rémi Watrigant.
URL:https://dimag.ibs.re.kr/event/2020-05-12/
LOCATION:Room B232\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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