We study the problem of finding large cuts in $d$-regular triangle-free graphs. In prior work, Shearer (1992) gives a randomised algorithm that finds a cut of expected size $(1/2 + 0.177/sqrt{d})m$, where $m$ is the number of edges. We give a simpler
algorithm that does much better: it finds a cut of expected size $(1/2 + 0.28125/sqrt{d})m$. As a corollary, this shows that in any $d$-regular triangle-free graph there exists a cut of at least this size. Our algorithm can be interpreted as a very efficient randomised distributed algorithm: each node needs to produce only one random bit, and the algorithm runs in one synchronous communication round. This work is also a case study of applying computational techniques in the design of distributed algorithms: our algorithm was designed by a computer program that searched for optimal algorithms for small values of $d$.
A local algorithm is a distributed algorithm that completes after a constant number of synchronous communication rounds. We present local approximation algorithms for the minimum dominating set problem and the maximum matching problem in 2-coloured a
nd weakly 2-coloured graphs. In a weakly 2-coloured graph, both problems admit a local algorithm with the approximation factor $(Delta+1)/2$, where $Delta$ is the maximum degree of the graph. We also give a matching lower bound proving that there is no local algorithm with a better approximation factor for either of these problems. Furthermore, we show that the stronger assumption of a 2-colouring does not help in the case of the dominating set problem, but there is a local approximation scheme for the maximum matching problem in 2-coloured graphs.
