  In the unsplittable flow problem on a path, we are given a capacitated path
$P$ and $n$ tasks, each task having a demand, a profit, and start and end
vertices. The goal is to compute a maximum profit set of tasks, such that for
each edge $e$ of $P$, the total demand of selected tasks that use $e$ does not
exceed the capacity of $e$. This is a well-studied problem that has been
studied under alternative names, such as resource allocation, bandwidth
allocation, resource constrained scheduling, temporal knapsack and interval
packing.
  We present a polynomial time constant-factor approximation algorithm for this
problem. This improves on the previous best known approximation ratio of
$O(\log n)$. The approximation ratio of our algorithm is $7+\epsilon$ for any
$\epsilon>0$.
  We introduce several novel algorithmic techniques, which might be of
independent interest: a framework which reduces the problem to instances with a
bounded range of capacities, and a new geometrically inspired dynamic program
which solves a special case of the maximum weight independent set of rectangles
problem to optimality. In the setting of resource augmentation, wherein the
capacities can be slightly violated, we give a $(2+\epsilon)$-approximation
algorithm. In addition, we show that the problem is strongly NP-hard even if
all edge capacities are equal and all demands are either~1,~2, or~3.
