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

Abstract

Given a graph $G=(V,E)$ with two distinguished vertices $s,t\in V$ and an integer $L$, an $L$-bounded flow is a flow between $s$ and $t$ that can be decomposed into paths of length at most $L$. In the 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. For networks with unit edge lengths (or, more generally, with polynomially bounded edge lengths, with respect to the number of vertices), the problem can be solved in polynomial time using linear programming. However, as far as we know, no polynomial-time combinatorial algorithm¹ for the $L$-bounded flow is known. For general edge lengths, the problem is NP-hard. 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 Říha in 1981. Unfortunately, their paper contains substantial 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 $(1+\varepsilon)$-approximation of the maximum $L$-bounded flow in time $\mathcal{O}(\varepsilon^{-2}m^2L\log L)$ 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. ¹Combinatorial in the sense that it does not explicitly use linear programming methods or methods from linear algebra or convex geometry.