Distance Constrained Vehicle Routing Problem to Minimize the Total Cost

Abstract

Given $\lambda >0$, an undirected complete graph $G=(V,E)$ with nonnegative edge-weight function obeying the triangle inequality and a depot vertex $r\in V$, a set $\{C_1,\ldots ,C_k\}$ of cycles is called a $\lambda$-bounded r-cycle cover if $V \subseteq \bigcup _{i=1}^k V(C_i)$ and each cycle $C_i$ contains r and has a length of at most $\lambda$. The Distance Constrained Vehicle Routing Problem with the objective of minimizing the total cost (DVRP-TC) aims to find a $\lambda$-bounded r-cycle cover $\{C_1,\ldots ,C_k\}$ such that the sum of the total length of the cycles and $\gamma k$ is minimized, where $\gamma$ is an input indicating the assignment cost of a single cycle.

For DVRP-TC on tree metric, we show a 2-approximation algorithm that is implied by the existing results and give an LP relaxation whose integrality gap has an upper bound of 5/2. In particular, when $\gamma =0$ we prove that this bound can be improved to 2. For the unrooted version of DVRP-TC, we devise a 5-approximation algorithm and show that a natural set-covering LP relaxation has a constant integrality gap of 25 using the rounding procedure given by Nagarajan and Ravi (2008).