Local algorithms for maximum cut and minimum bisection on locally treelike regular graphs of large degree

Given a graph G$$ G $$ of degree k$$ k $$ over n$$ n $$ vertices, we consider the problem of computing a near maximum cut or a near minimum bisection in polynomial time. For graphs of girth 2L$$ 2L $$, we develop a local message passing algorithm whose complexity is O(nkL)$$ O(nkL) $$, and that achi...

Celý popis

Uložené v:
Podrobná bibliografia
Vydané v:Random structures & algorithms Ročník 63; číslo 3; s. 689 - 715
Hlavní autori: El Alaoui, Ahmed, Montanari, Andrea, Sellke, Mark
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: New York John Wiley & Sons, Inc 01.10.2023
Wiley Subscription Services, Inc
Predmet:
Algorithms
Apexes
Combinatorial analysis
graph partionning
Graph theory
Graphs
local algorithms
maximum cut
Message passing
minimum bisection
Polynomials
Spin glasses
ISSN:1042-9832, 1098-2418
On-line prístup:Získať plný text
Tagy: Pridať tag
Žiadne tagy, Buďte prvý, kto otaguje tento záznam!
Popis
Shrnutí:Given a graph G$$ G $$ of degree k$$ k $$ over n$$ n $$ vertices, we consider the problem of computing a near maximum cut or a near minimum bisection in polynomial time. For graphs of girth 2L$$ 2L $$, we develop a local message passing algorithm whose complexity is O(nkL)$$ O(nkL) $$, and that achieves near optimal cut values among all L$$ L $$‐local algorithms. Focusing on max‐cut, the algorithm constructs a cut of value nk/4+nP⋆k/4+err(n,k,L)$$ nk/4+n{P}_{\star}\sqrt{k/4}+\mathsf{err}\left(n,k,L\right) $$, where P⋆≈0.763166$$ {P}_{\star}\approx 0.763166 $$ is the value of the Parisi formula from spin glass theory, and err(n,k,L)=on(n)+nok(k)+nkoL(1)$$ \mathsf{err}\left(n,k,L\right)={o}_n(n)+n{o}_k\left(\sqrt{k}\right)+n\sqrt{k}{o}_L(1) $$ (subscripts indicate the asymptotic variables). Our result generalizes to locally treelike graphs, that is, graphs whose girth becomes 2L$$ 2L $$ after removing a small fraction of vertices. Earlier work established that, for random k$$ k $$‐regular graphs, the typical max‐cut value is nk/4+nP⋆k/4+on(n)+nok(k)$$ nk/4+n{P}_{\star}\sqrt{k/4}+{o}_n(n)+n{o}_k\left(\sqrt{k}\right) $$. Therefore our algorithm is nearly optimal on such graphs. An immediate corollary of this result is that random regular graphs have nearly minimum max‐cut, and nearly maximum min‐bisection among all regular locally treelike graphs. This can be viewed as a combinatorial version of the near‐Ramanujan property of random regular graphs.
Bibliografia:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:1042-9832
1098-2418
DOI:10.1002/rsa.21149