Euclidean Capacitated Vehicle Routing in Random Setting: A 1.55-Approximation Algorithm
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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 ε>0, our algorithm is a (1+ε)-approximation asymptotically almost surely.
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