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...

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Bibliographic Details
Published in:Random structures & algorithms Vol. 63; no. 3; pp. 689 - 715
Main Authors: El Alaoui, Ahmed, Montanari, Andrea, Sellke, Mark
Format: Journal Article
Language:English
Published: New York John Wiley & Sons, Inc 01.10.2023
Wiley Subscription Services, Inc
Subjects:
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
Online Access:Get full text
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Summary: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.
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ISSN:1042-9832
1098-2418
DOI:10.1002/rsa.21149