Algorithms for Interval Coloring with Capacities and Demands
If the demands satisfy the no-bottleneck assumption, that is $\max_i d_i \leq \min_e c(e)$ then a constant factor approximation is known even for trees. See http://dl.acm.org/citation.cfm?id=1273343 which gives a 48-approx for trees. For paths a better bound can be obtained and may have been shown but I am not sure where or whether it was published. For the general case it is not quite clear what the best known result is. One can get an easy $O(\log n)$-approximation as follows. Consider the problem of maximizing the number of requests that can be feasibly routed; there is a constant factor approximation for this problem, see http://arxiv.org/abs/1102.3643. Using this as a black box one can do a greedy set-cover like algorithm to repeatedly pack as many requests as possible to get the desired $O(\log n)$-approximation. A paper by Chalermsook on coloring rectangles may have some implications including giving an $O(\log \log n)$-approximation; the paper is available at http://sites.google.com/site/chalermsook/academic/coloring_full.pdf?attredirects=0. However it may require figuring out several technical details. One suspects that there is a constant factor approximation for the coloring problem.