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A unified view of graph regularity via matrix decompositions
Author(s) -
Bodwin Greg,
Vempala Santosh
Publication year - 2022
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.21053
Subject(s) - mathematics , combinatorics , indifference graph , pseudorandomness , discrete mathematics , modular decomposition , dense graph , pathwidth , line graph , graph , pseudorandom number generator , algorithm
We give a unified proof of algorithmic weak and Szemerédi regularity lemmas for several well‐studied classes of sparse graphs, for which only weak regularity lemmas were previously known. These include core‐dense graphs, low threshold rank graphs, and (a version of)L pupper regular graphs. More precisely, we define cut pseudorandom graphs , we prove our regularity lemmas for these graphs, and then we show that cut pseudorandomness captures all of the above graph classes as special cases. The core of our approach is an abstracted matrix decomposition, which can be computed by a simple algorithm by Charikar. Using work of Oveis Gharan and Trevisan, it also implies new PTASes for MAX‐CUT, MAX‐BISECTION, MIN‐BISECTION for a significantly expanded class of input graphs. (It is NP Hard to get PTASes for these graphs in general.)