Approximation Algorithms for Movement Repairmen

In the {\em Movement Repairmen (MR)} problem we are given a metric space $(V, d)$ along with a set $R$ of $k$ repairmen $r_1, r_2, ..., r_k$ with their start depots $s_1, s_2, ..., s_k \in V$ and speeds $v_1, v_2, ..., v_k \geq 0$ respectively and a set $C$ of $m$ clients $c_1, c_2, ..., c_m$ having start locations $s'_1, s'_2, ..., s'_m \in V$ and speeds $v'_1, v'_2, ..., v'_m \geq 0$ respectively. If $t$ is the earliest time a client $c_j$ is collocated with any repairman (say, $r_i$) at a node $u$, we say that the client is served by $r_i$ at $u$ and that its latency is $t$. The objective in the (\smr{}) problem is to plan the movements for all repairmen and clients to minimize the sum (average) of the clients latencies. The motivation for this problem comes, for example, from Amazon Locker Delivery \cite{amazon} and USPS gopost \cite{gopost}. We give the first $O(\log n)$-approximation algorithm for the \smr{} problem.