Approximation Algorithms for Regret-Bounded Vehicle Routing and Applications to Distance-Constrained Vehicle Routing

We consider vehicle-routing problems (VRPs) that incorporate the notion of {\em regret} of a client, which is a measure of the waiting time of a client relative to its shortest-path distance from the depot. Formally, we consider both the additive and multiplicative versions of, what we call, the {\em regret-bounded vehicle routing problem} (RVRP). In these problems, we are given an undirected complete graph $G=(\{r\}\cup V,E)$ on $n$ nodes with a distinguished root (depot) node $r$, edge costs $\{c_{uv}\}$ that form a metric, and a regret bound $R$. Given a path $P$ rooted at $r$ and a node $v\in P$, let $c_P(v)$ be the distance from $r$ to $v$ along $P$. The goal is to find the fewest number of paths rooted at $r$ that cover all the nodes so that for every node $v$ covered by (say) path $P$: (i) its additive regret $c_P(v)-c_{rv}$, with respect to $P$ is at most $R$ in {\em additive-RVRP}; or (ii) its multiplicative regret, $c_P(c)/c_{rv}$, with respect to $P$ is at most $R$ in {\em multiplicative-RVRP}. Our main result is the {\em first} constant-factor approximation algorithm for additive-RVRP by devising rounding techniques for a natural {\em configuration-style LP}. This is a substantial improvement over the previous-best $O(\log n)$-approximation. Additive-RVRP turns out be a rather central vehicle-routing problem, whose study reveals insights into a variety of other regret-related problems as well as the classical {\em distance-constrained VRP} ({DVRP}). We obtain approximation ratios of $O\bigl(\log(\frac{R}{R-1})\bigr)$ for multiplicative-RVRP, and $O\bigl(\min\bigl\{\mathit{OPT},\frac{\log D}{\log\log D}\bigr\}\bigr)$ for DVRP with distance bound $D$ via reductions to additive-RVRP; the latter improves upon the previous-best approximation for DVRP.