Given a graph $G=(V,E)$ with two distinguished vertices $s,tin V$ 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 $O(epsilon^{-2}m^2 Llog L)$ 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.
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