We are given a graph $G = (V,E)$, where each edge $e \in E$ has a capacity $c_e$. There are a set of $k$ requests $R_1,\ldots,R_k$. Request $R_i$ has a demand $d_i$, and has an interval $I_i = [s_i,t_i]$ associated with it, where $s_i$ is the source and $t_i$ is the destination vertex. A set of requests $\mathcal{R}$ is called feasible, if the total demand of all requests in $\mathcal{R}$ passing through any edge $e$ does not exceed it's capacity, i.e., $\sum_{i:e \in R_i} d_i \le c_e$ for all $e \in E$. The goal is to partition the requests $R_1,\ldots,R_k$ into a number of sets such that each set is feasible and the total number of sets is minimized. This problem is called Interval Coloring with Capacities and Demands. This is because we can think of this as coloring the requests in such a manner so that requests in each color class are feasible and we want to minimize the number of colors. This can also be thought of as routing the requests in a feasible manner in a number of rounds. We consider the special case where $G$ is a path $(v_1,e_1,v_2,e_2,\ldots,e_{n-1},v_n)$.
What is the best approximation algorithm and hardness result known for this problem?