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# Eun Jung Kim (김은정), Twin-width: tractable FO model checking

## Tuesday, May 12, 2020 @ 4:30 PM - 5:30 PM KST

Room B232, IBS (기초과학연구원)

### Speaker

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].

In 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.

Joint work with Stéphan Thomassé, Édouard Bonnet, and Rémi Watrigant.

## Details

Date:
Tuesday, May 12, 2020
Time:
4:30 PM - 5:30 PM KST
Event Category:
Event Tags:
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Room B232
IBS (기초과학연구원)

## Organizer

Sang-il Oum (엄상일)
View Organizer Website
기초과학연구원 수리및계산과학연구단 이산수학그룹
대전 유성구 엑스포로 55 (우) 34126
IBS Discrete Mathematics Group (DIMAG)
Institute for Basic Science (IBS)
55 Expo-ro Yuseong-gu Daejeon 34126 South Korea
E-mail: dimag@ibs.re.kr, Fax: +82-42-878-9209