René Sitters

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 $n^{f(ε)\cdot\log\log n}$ for any $c$, and where $f$ is a function of $ε$ only. This is a major improvement over the so far best known running time of $n^{\log^{O(1/ε)}n}$ 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.

We introduce the Chinese deliveryman problem where the goal of the deliveryman is to visit every house in his neighborhood such that the average time of arrival is minimized. We show that, in contrast to the well-known Chinese postman problem, the Chinese deliveryman problem is APX-hard in general and NP-hard for planar graphs. We give a simple $$\sqrt{2}$$2 -approximation for undirected graphs and a 4 / 3-approximation for 2-edge connected graphs. We observe that there is a PTAS for planar graphs and that depth first search is optimal for trees.