An $n^{O(\log\log n)}$ time approximation scheme for capacitated VRP in the Euclidean plane

We present a quasi polynomial time approximation scheme (Q-PTAS) for the capacitated vehicle routing problem (CVRP) on $n$ points in the Euclidean plane for arbitrary capacity $c$. The running time is $n^{f(\epsilon)\cdot\log\log n}$ for any $c$, and where $f$ is a function of $\epsilon$ only. This is a major improvement over the so far best known running time of $n^{\log^{O(1/\epsilon)}n}$ time and a big step towards a PTAS for Euclidean CVRP. In our algorithm, we first give a polynomial time reduction of the CVRP in $\mathbb{R}^d$ (for any fixed $d$) to an uncapacitated routing problem in $\mathbb{R}^d$ that we call the $m$-paths problem. Here, one needs to find exactly $m$ paths between two points $a$ and $b$, covering all the given points in the Euclidean space. We then give a Q-PTAS for the $m$-paths problem in the pane. Any PTAS for the (arguably easier to handle) Euclidean $m$-paths problem is most likely to imply a PTAS for the Euclidean CVRP.