  We define supersymmetric Yang-Mills theory on an arbitrary two-dimensional
lattice (polygon decomposition) with preserving one supercharge. When a smooth
Riemann surface $\Sigma_g$ with genus $g$ emerges as an appropriate continuum
limit of the generic lattice, the discretized theory becomes topologically
twisted $\mathcal{N}=(2,2)$ supersymmetric Yang-Mills theory on $\Sigma_g$. If
we adopt the usual square lattice as a special case of the discretization, our
formulation is identical with Sugino's lattice model. Although the tuning of
parameters is generally required while taking the continuum limit, the number
of the necessary parameters is at most two because of the gauge symmetry and
the supersymmetry. In particular, we do not need any fine-tuning if we arrange
the theory so as to possess an extra global U(1) symmetry ($U(1)_{R}$ symmetry)
which rotates the scalar fields.
