The capacitated vehicle routing problem (c-VRP), which is also known as the k-tour cover problem, has been widely studied in many settings. In the Euclidean plane, a PTAS for constant capacity c was given more than 40 years ago, which has been improved and extended ever since. Here, we present arguably the easiest form of a PTAS possible. The algorithm simply chooses between greedy or complete enumeration. A direct corollary of this simplification is that the PTAS extends to basically any kind of order restriction, as long as the objective is to minimize total tour length. Furthermore, we show that when the metric is a tree, there is a clear separation in complexity between the c-VRP with and without order restrictions. We show that even for a fixed order, the c-VRP is APX-hard for arbitrary (non-constant) capacity c. This in contrast to the polynomial time approximation scheme for the standard c-VRP on trees.

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On the Complexity of Capacitated Vehicle Routing with Order Restrictions

  • Steven Miltenburg,
  • Tim Oosterwijk,
  • René Sitters

摘要

The capacitated vehicle routing problem (c-VRP), which is also known as the k-tour cover problem, has been widely studied in many settings. In the Euclidean plane, a PTAS for constant capacity c was given more than 40 years ago, which has been improved and extended ever since. Here, we present arguably the easiest form of a PTAS possible. The algorithm simply chooses between greedy or complete enumeration. A direct corollary of this simplification is that the PTAS extends to basically any kind of order restriction, as long as the objective is to minimize total tour length. Furthermore, we show that when the metric is a tree, there is a clear separation in complexity between the c-VRP with and without order restrictions. We show that even for a fixed order, the c-VRP is APX-hard for arbitrary (non-constant) capacity c. This in contrast to the polynomial time approximation scheme for the standard c-VRP on trees.