  Given a graph $G = (V, E)$, the $3$-path partition problem is to find a
minimum collection of vertex-disjoint paths each of order at most $3$ to cover
all the vertices of $V$. It is different from but closely related to the
well-known $3$-set cover problem. The best known approximation algorithm for
the $3$-path partition problem was proposed recently and has a ratio $13/9$.
Here we present a local search algorithm and show, by an amortized analysis,
that it is a $4/3$-approximation. This ratio matches up to the best
approximation ratio for the $3$-set cover problem.
