  We study the unit-demand capacitated vehicle routing problem in the random
setting of the Euclidean plane. The objective is to visit $n$ random terminals
in a square using a set of tours of minimum total length, such that each tour
visits the depot and at most $k$ terminals.
  We design an elegant algorithm combining the classical sweep heuristic and
Arora's framework for the Euclidean traveling salesman problem [Journal of the
ACM 1998]. We show that our algorithm is a polynomial-time approximation of
ratio at most $1.55$ asymptotically almost surely. This improves on previous
approximation ratios of $1.995$ due to Bompadre, Dror, and Orlin [Journal of
Applied Probability 2007] and $1.915$ due to Mathieu and Zhou [Random
Structures and Algorithms 2022]. In addition, we conjecture that, for any
$\varepsilon>0$, our algorithm is a $(1+\varepsilon)$-approximation
asymptotically almost surely.
