Large cuts with local algorithms on triangle-free graphs

Juho Hirvonen¹ ; Joel Rybicki² ; Stefan Schmid³ ; Jukka Suomela⁴

¹IRIF, CNRS and University Paris Diderot
²Department of Biosciences, University of Helsinki
³Department of Computer Science, Aalborg University
⁴Department of Computer Science, Aalto University

The electronic journal of combinatorics, Tome 24 (2017) no. 4

Résumé

Let $G$ be a $d$-regular triangle-free graph with $m$ edges. We present an algorithm which finds a cut in $G$ with at least $(1/2 + 0.28125/\sqrt{d})m$ 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 $\mathcal{N}_d$ such that there is a one-to-one correspondence between heavy cuts of $\mathcal{N}_d$ 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 $\mathcal{N}_d$ to attain a lower bound on the maximum cut size of any $d$-regular triangle-free graph.

Détail

DOI : 10.37236/6862

Classification : 05C85
Mots-clés : graph theory, regular graphs, cuts