In this paper, we propose a novel approach to solving the distributed constraint optimisation problem (DCOP) that guarantees completeness, while having linear communication complexity. The key to performance advantages, in terms of both computation and communication, derives from the application of the repeatedly-half principle to manage complexity by a combination of problem distribution through graph decomposition and multi-stage solution quality propagation. Experimental result shows that our new algorithm is faster than a recent competitive distributed algorithm for solving MaxSAT graph colouring problems. It also indicates the potential for the decomposition approach over a centralised method based on the same search strategy, and is consistent with recent results on domain propagation and structural decomposition in the CSP literature.