# On Polynomial-Time Combinatorial Algorithms for Maximum $L$-Bounded Flow

## TL;DR

This paper introduces a new combinatorial approximation algorithm for the maximum L-bounded flow problem, providing the first known polynomial-time combinatorial solution with provable approximation guarantees.

## Contribution

It presents a novel combinatorial algorithm based on the exponential length method for approximating maximum L-bounded flow, addressing the lack of such algorithms in prior work.

## Key findings

- Provides a $(1+psilon)$-approximation algorithm with runtime $O(psilon^{-2}m^2 L \u221alog L)$
- Shows the approach works for NP-hard generalizations with edge lengths
- Identifies flaws in previous attempts at combinatorial algorithms for the problem

## Abstract

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.