  The scaling behavior of a directed polymer in a two-dimensional (2D) random
potential under confining force is investigated. The energy of a polymer with
configuration $\{y(x)\}$ is given by $H\big(\{y(x)\}\big) = \sum_{x=1}^N \exyx
+ \epsilon \Wa^\alpha$, where $\eta(x,y)$ is an uncorrelated random potential
and $\Wa$ is the width of the polymer. Using an energy argument, it is
conjectured that the radius of gyration $R_g(N)$ and the energy fluctuation
$\Delta E(N)$ of the polymer of length $N$ in the ground state increase as
$R_g(N)\sim N^{\nu}$ and $\Delta E(N)\sim N^\omega$ respectively with $\nu =
1/(1+\alpha)$ and $\omega = (1+2\alpha)/(4+4\alpha)$ for $\alpha\ge 1/2$. A
novel algorithm of finding the exact ground state, with the effective time
complexity of $\cO(N^3)$, is introduced and used to confirm the conjecture
numerically.
