  We study the annihilating random walk with long-range interaction in one
dimension. Each particle performs random walks on a one-dimensional ring in
such a way that the probability of hopping toward the nearest particle is $W=
[1 - \epsilon (x+\mu)^{-\sigma}]/2$ (the probability of moving away from its
nearest particle is $1-W$), where $x$ is the distance from the hopping particle
to its nearest particle and $\epsilon$, $\mu$, and $\sigma$ are parameters. For
positive (negative) $\epsilon$, a particle is effectively repulsed (attracted)
by its nearest particle and each hopping is generally biased. On encounter, two
particles are immediately removed from the system. We first study the survival
probability and the mean spreading behaves in the long-time limit if there are
only two particles in the beginning. Then, we study how the density decays to
zero if all sites are occupied at the outset. We find that the asymptotic
behaviors are classified by seven categories: (i) $\sigma>1$ or $\epsilon=0$,
(ii) $\sigma = 1$ and $2\epsilon > 1$, (iii) $\sigma=1$ and $2\epsilon = 1$,
(iv) $\sigma = 1$ and $2\epsilon < 1$, (v) $\sigma<1$ and $\epsilon > 0$, (vi)
$\sigma = 0$ and $\epsilon<0$, and (vii) $0 < \sigma <1$ and $\epsilon<0$. The
asymptotic behaviors in each category are universal in the sense that $\mu$
(and sometimes $\epsilon$) cannot affect the asymptotic behaviors.
