$NP$-Completeness of $\epsilon$-balanced graph partitioning for fixed $\epsilon$
The case where $\epsilon$ is a fixed constant in $(0,1)$ represents a partition that each part contains at least a constant fraction of the vertices. This was proven to be NP-complete by Wagner and Wagner. This is also known as balanced minimum cut problem. Note that $|V_i| \leq (1+\epsilon)|V|/2$ if and only if $|V_i| \ge (1-\epsilon)|V|/2$ for $\epsilon$ in $(0,1)$.

Wagner D., Wagner F. (1993) Between Min Cut and Graph Bisection. In: Borzyszkowski A.M., Sokołowski S. (eds) Mathematical Foundations of Computer Science 1993. MFCS 1993. Lecture Notes in Computer Science, vol 711. Springer, Berlin, Heidelberg