Iterated tour partitioning for Euclidean capacitated vehicle routing

We give a probabilistic analysis of the unit-demand Euclidean capacitated vehicle routing problem in the random setting. The objective is to visit all customers using a set of routes of minimum total length, such that each route visits at most k$$ k $$ customers. The best known polynomial-time approximation is the iterated tour partitioning (ITP) algorithm, introduced in 1985 by Haimovich and Rinnooy Kan. They showed that the solution obtained by the ITP algorithm is arbitrarily close to the optimum when k$$ k $$ is either o(n)$$ o\left(\sqrt{n}\right) $$ or omega(n)$$ \omega \left(\sqrt{n}\right) $$, and they asked whether the ITP algorithm was "also effective in the intermediate range". In this work, we show that the ITP algorithm is at best a (1+c0)$$ \left(1+{c}_0\right) $$-approximation, for some positive constant c0$$ {c}_0 $$, and is at worst a 1.915-approximation.