Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Polynomial-Time Combinatorial Algorithms for Maximum $L$-Bounded Flow   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.

26
The categories of this paper are: bayes-an
17
The categories of this paper are: q-fin
14
The categories of this paper are: math-ph
19
The categories of this paper are: eess
37
The categories of this paper are: supr-con
2
The categories of this paper are: math
21
The categories of this paper are: adap-org
16
The categories of this paper are: stat
20
The categories of this paper are: acc-phys
3
The categories of this paper are: physics
10
The categories of this paper are: q-bio
4
The categories of this paper are: cond-mat
6
The categories of this paper are: astro-ph
24
The categories of this paper are: dg-ga
15
The categories of this paper are: nucl-ex
34
The categories of this paper are: chem-ph