Large Cuts with Local Algorithms on Triangle-Free Graphs

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Hirvonen , J , Rybicki , J , Schmid , S & Suomela , J 2017 , ' Large Cuts with Local Algorithms on Triangle-Free Graphs ' , The Electronic Journal of Combinatorics , vol. 24 , no. 4 , ARTN P4.21 .

Title: Large Cuts with Local Algorithms on Triangle-Free Graphs
Author: Hirvonen, Juho; Rybicki, Joel; Schmid, Stefan; Suomela, Jukka
Contributor: University of Helsinki, CNRS, Centre National de la Recherche Scientifique (CNRS), IRIF
University of Helsinki, Biosciences
University of Helsinki, Aalto University
Date: 2017-10-20
Language: eng
Number of pages: 20
Belongs to series: The Electronic Journal of Combinatorics
ISSN: 1077-8926
Abstract: Let G be a d-regular triangle-free graph with in edges. We present an algorithm which finds a cut in G with at least (1/2 + 0.28125/root d)rn edges in expectation, improving upon Shearer's classic result. In particular, this implies that any d-regular triangle-free graph has a cut of at least this size, and thus, we obtain a new lower bound for the maximum number of edges in a bipartite subgraph of G. Our algorithm is simpler than Shearer's classic algorithm and it can be interpreted as a very efficient randomised distributed (local) algorithm: each node needs to produce only one random bit, and the algorithm runs in one round. The randomised algorithm itself was discovered using computational techniques. We show that for any fixed d, there exists a weighted neighbourhood graph N-d such that there is a one-to-one correspondence between heavy cuts of N-d and randomised local algorithms that find large cuts in any d-regular input graph. This turns out to be a useful tool for analysing the existence of cuts in d-regular graphs: we can compute the optimal cut of N-d to attain a lower bound on the maximum cut size of any d-regular triangle-free graph.
Subject: graph theory
regular graphs
111 Mathematics

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