Large Cuts with Local Algorithms on Triangle-Free Graphs

Research output: Contribution to journalArticle

Details

Original languageEnglish
Article numberP4.21
Pages (from-to)1-20
Number of pages20
JournalELECTRONIC JOURNAL OF COMBINATORICS
Volume24
Issue number4
StatePublished - 20 Oct 2017
MoE publication typeA1 Journal article-refereed

Researchers

Research units

  • CNRS - IRIF
  • CNRS
  • Aalborg University
  • University Paris Diderot
  • University of Helsinki

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.

    Research areas

  • graph theory, regular graphs, cuts, BIPARTITE SUBGRAPHS, RAMANUJAN GRAPHS, MAX-CUT, APPROXIMATION

ID: 16140772