A Constant Factor Approximation Algorithm for Unsplittable Flow on Paths

TL;DR

This paper introduces a polynomial-time constant-factor approximation algorithm for the unsplittable flow problem on paths, significantly improving previous approximation ratios and employing novel algorithmic techniques.

Contribution

It presents the first polynomial-time constant-factor approximation algorithm with ratio $7+\, ext{ extepsilon}$ for the problem, surpassing the prior $O(\, extlog n)$ ratio, and introduces new algorithmic frameworks.

Findings

Achieves a $7+ extepsilon$ approximation ratio.

Provides a $(2+ extepsilon)$-approximation under resource augmentation.

Proves the problem is strongly NP-hard even with uniform capacities and small demands.

Abstract

In the unsplittable flow problem on a path, we are given a capacitated path $P$ and $n$ tasks, each task having a demand, a profit, and start and end vertices. The goal is to compute a maximum profit set of tasks, such that for each edge $e$ of $P$, the total demand of selected tasks that use $e$ does not exceed the capacity of $e$. This is a well-studied problem that has been studied under alternative names, such as resource allocation, bandwidth allocation, resource constrained scheduling, temporal knapsack and interval packing. We present a polynomial time constant-factor approximation algorithm for this problem. This improves on the previous best known approximation ratio of $O(\log n)$. The approximation ratio of our algorithm is $7+\epsilon$ for any $\epsilon>0$. We introduce several novel algorithmic techniques, which might be of independent interest: a framework which reduces the problem to instances with a bounded range of capacities, and a new geometrically inspired dynamic program which solves a special case of the maximum weight independent set of rectangles problem to optimality. In the setting of resource augmentation, wherein the capacities can be slightly violated, we give a $(2+\epsilon)$-approximation algorithm. In addition, we show that the problem is strongly NP-hard even if all edge capacities are equal and all demands are either~1,~2, or~3.