# Yunbum Kook (국윤범), Vertex Sparsification for Edge Connectivity

## August 11 Tuesday @ 4:30 PM - 5:30 PM KST

Room B232, IBS (기초과학연구원)

### Speaker

Yunbum Kook (국윤범)
Department of Mathematical Sciences, KAIST
https://yunbum-kook.github.io/

Graph compression or sparsification is a basic information-theoretic and computational question. A major open problem in this research area is whether $(1+\epsilon)$-approximate cut-preserving vertex sparsifiers with size close to the number of terminals exist. As a step towards this goal, we initiate the study of a thresholded version of the problem: for a given parameter $c$, find a smaller graph, which we call connectivity-$c$ mimicking network, which preserves connectivity among $k$ terminals exactly up to the value of $c$. We show that contraction-based connectivity-$c$ mimicking networks with $O(kc^4)$ edges exist by (1) introducing an extension of well-linkedness to a thresholded $c$-connectivity setting and (2) leveraging a kernelization result, based on gammoid and the representative sets lemma, to identify `essential edges’ in minimum edge cuts between a partition of terminals. We also develop an algorithm based on expander decomposition, which can find a contraction-based $c$-mimicking network of the optimal size in $m(c\log n)^{O(c)}$.

These results lead to the first data structures for answering fully dynamic offline $c$-edge-connectivity queries for $c \ge 4$ in polylogarithmic time per query, as well as more efficient algorithms for survivable network design on bounded treewidth graphs.

This is a joint work with Parinya Chalermsook, Syamantak Das, Bundit Laekhanukit, Yang P. Liu, Richard Peng, Mark Sellke, and Daniel Vaz.

## Details

Date:
August 11 Tuesday
Time:
4:30 PM - 5:30 PM KST
Event Category:
Event Tags:
,

Room B232
IBS (기초과학연구원)

## Organizer

Sang-il Oum (엄상일)
Website:
https://dimag.ibs.re.kr/home/sangil/
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