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We prove a multivariate central limit theorem with explicit error bound on anon-smooth function distance for sums of bounded decomposable $d$-dimensionalrandom vectors. The decomposition structure is similar to that of Barbour,Karo\'nski and Ruci\'nski (1989) and is more general than the local dependencestructure considered in Chen and Shao (2004). The error bound is of the order$d^{\frac{1}{4}} n^{-\frac{1}{2}}$, where $d$ is the dimension and $n$ is thenumber of summands. The dependence on $d$, namely $d^{\frac{1}{4}}$, is thebest known dependence even for sums of independent and identically distributedrandom vectors, and the dependence on $n$, namely $n^{-\frac{1}{2}}$, isoptimal. We apply our main result to a random graph example.

A multivariate CLT for bounded decomposable random vectors with the best known rate