Servicing Timed Requests on a Line

We consider an off-line optimisation problem where $k$ robots must service $n$ requests on a single line. A request $i$ has weight $w_i$ and takes place at time $t_i$ at location $d_i$ on the line. A robot can service a request and collect the weight $w_i$, if it is present at $d_i$ at time $t_i$. The objective is to find $k$ robot-schedules that maximize the total weight. The optimisation problem is motivated by a robotics application [Asahiro et al. Discrete Applied Mathematics, 2006] and can be modeled as a minimum cost flow problem with unit capacities in a flow network $\mathcal{N}$. Consequently, we ask for a collection of $k$ node-disjoint paths from the source $s$ to the sink $t$ in $\mathcal{N}$, with minimum total weight. It was shown in [Asahiro et al. Discrete Applied Mathematics, 2006] that the flow network $\mathcal{N}$ can be implicitly represented by $n$ points on the plane which yields to an $O(n \log n)$-time algorithm for $k=1$ and the special case where all requests have the same weight. However, for $k \geq 2$ the problem can be solved in $O(kn^2)$ time with the successive shortest path algorithm which does not use this implicit representation. We consider arbitrary request weights and show a recursive $O(k^{2k}n \log^{2k} n)$-time algorithm which improves the previous bound if $k$ is considered constant. Our result also improves the running time of previous algorithms for other variants of the optimisation problem. Finally, we show problem properties that may be useful within the context of applications that motivate the problem and may yield to more efficient algorithms.