Probabilistic Analysis of Euclidean Capacitated Vehicle Routing
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We give a probabilistic analysis of the unit-demand Euclidean capacitated vehicle routing problem in the random setting, where the input distribution consists of n unit-demand customers modeled as independent, identically distributed uniform random points in the two-dimensional plane. The objective is to visit every customer using a set of routes of minimum total length, such that each route visits at most k customers, where k is the capacity of a vehicle. All of the following results are in the random setting and hold asymptotically almost surely. The best known polynomial-time approximation for this problem is the iterated tour partitioning (ITP) algorithm, introduced in 1985 by Haimovich and Rinnooy Kan. They showed that the ITP algorithm is near-optimal when k is either o(√(n)) or ω(√(n)), and they asked whether the ITP algorithm was also effective in the intermediate range. In this work, we show that when k=√(n), the ITP algorithm is at best a (1+c_0)-approximation for some positive constant c_0. On the other hand, the approximation ratio of the ITP algorithm was known to be at most 0.995+α due to Bompadre, Dror, and Orlin, where α is the approximation ratio of an algorithm for the traveling salesman problem. In this work, we improve the upper bound on the approximation ratio of the ITP algorithm to 0.915+α. Our analysis is based on a new lower bound on the optimal cost for the metric capacitated vehicle routing problem, which may be of independent interest.
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