  Given a graph $G=(V,E)$ with two distinguished vertices $s,t\in V$ and an
integer $L$, an {\em $L$-bounded flow} is a flow between $s$ and $t$ that can
be decomposed into paths of length at most $L$. In the {\em maximum $L$-bounded
flow problem} the task is to find a maximum $L$-bounded flow between a given
pair of vertices in the input graph.
  The problem can be solved in polynomial time using linear programming.
However, as far as we know, no polynomial-time combinatorial algorithm for the
$L$-bounded flow is known. The only attempt, that we are aware of, to describe
a combinatorial algorithm for the maximum $L$-bounded flow problem was done by
Koubek and \v{R}\'i ha in 1981. Unfortunately, their paper contains
substantional flaws and the algorithm does not work; in the first part of this
paper, we describe these problems.
  In the second part of this paper we describe a combinatorial algorithm based
on the exponential length method that finds a $(1+\epsilon)$-approximation of
the maximum $L$-bounded flow in time $O(\epsilon^{-2}m^2 L\log L)$ where $m$ is
the number of edges in the graph. Moreover, we show that this approach works
even for the NP-hard generalization of the maximum $L$-bounded flow problem in
which each edge has a length.
