  Given a directed acyclic graph with positive edge-weights, two vertices s and
t, and a threshold-weight L, we present a fully-polynomial time
approximation-scheme for the problem of counting the s-t paths of length at
most L. We extend the algorithm for the case of two (or more) instances of the
same problem. That is, given two graphs that have the same vertices and edges
and differ only in edge-weights, and given two threshold-weights L_1 and L_2,
we show how to approximately count the s-t paths that have length at most L_1
in the first graph and length at most L_2 in the second graph. We believe that
our algorithms should find application in counting approximate solutions of
related optimization problems, where finding an (optimum) solution can be
reduced to the computation of a shortest path in a purpose-built auxiliary
graph.
