Reverse discrepancy and almost zero-sum stars

For $f$ chosen from the $\{-1,1\}$-valued functions on the edges of a hypergraph $\mathcal{H}=(V,E)$ with ${\sum }_{e\in E}f(e)=0$, how large can one make ${\min }_{v\in V}\mathrm{\mid }{\sum }_{e\ni v}f(e)\mathrm{\mid }$? This question may be viewed as a reverse version of the hypergraph discrepancy problem or as a relaxation of the zero-sum Ramsey problem for stars. We prove exact results when $\mathcal{H}$ is a complete or equipartite hypergraph.