A Mazing 2+ε Approximation for Unsplittable Flow on a Path

Abstract

We study the problem of unsplittable flow on a path (UFP), which arises naturally in many applications such as bandwidth allocation, job scheduling, and caching. Here we are given a path with nonnegative edge capacities and a set of tasks, which are characterized by a subpath, a demand, and a profit. The goal is to find the most profitable subset of tasks whose total demand does not violate the edge capacities. Not surprisingly, this problem has received a lot of attention in the research community. If the demand of each task is at most a small-enough fraction δ of the capacity along its subpath (δ- small tasks ), then it has been known for a long time [Chekuri et al., ICALP 2003] how to compute a solution of value arbitrarily close to the optimum via LP rounding. However, much remains unknown for the complementary case, that is, when the demand of each task is at least some fraction δ > 0 of the smallest capacity of its subpath (δ- large tasks ). For this setting, a constant factor approximation is known, improving on an earlier logarithmic approximation [Bonsma et al., FOCS 2011]. In this article, we present a polynomial-time approximation scheme (PTAS) for δ-large tasks, for any constant δ > 0. Key to this result is a complex geometrically inspired dynamic program. Each task is represented as a segment underneath the capacity curve, and we identify a proper maze-like structure so that each corridor of the maze is crossed by only O (1) tasks in the optimal solution. The maze has a tree topology, which guides our dynamic program. Our result implies a 2+ε approximation for UFP, for any constant ε > 0, improving on the previously best 7+ε approximation by Bonsma et al. We remark that our improved approximation algorithm matches the best known approximation ratio for the considerably easier special case of uniform edge capacities.

To Augment or Not to Augment: Solving Unsplittable Flow on a Path by Creating Slack

“…This generalizes (and improves) previous results for the cases of uniform edge capacities [11] and the no-bottleneck-assumption (NBA) [13] (the NBA requires that max i∈T d(i) ≤ min e∈E u(e)). Since relatively large tasks are easy to handle, O(1)-and (2+ε)-approximation algorithms were known for these two cases [11,13,16] before they were known for the general case [3,10].…”

“…the rightmost edge to the left of e k−1 ) such thatδ e k <δ e k−1 . Consider a task i of kind (3). In one of the two sequences above, there exists some k ≥ 1 such thatδ e k ≤ d(i) <δ e k−1 .…”

“…To achieve that, Bansal et al reduced the general case to the rooted case of the problem and provided an O(1)-approximation for the latter. After an improvement to (7 + ε)-approximation by Bonsma et al [10], the current best approximation ratio is 2 + ε due to Anagnostopoulos et al [3].…”

“…All previous polynomial time approximation algorithms for the problem and its special cases [3,10,11,13,16] essentially treat these two groups separately, which inherently lose a factor of 2 in the approximation ratio.…”

Dedicated Protection for Survivable Virtual Network Embedding

“…State-of-the art literature reveals that this problem is also very hard to approximate even when the source and destination of the flows are known. Recent research works have found (2 + ε) approximation algorithms for line and cycle graphs, respectively [19]. However, finding constant factor approximation algorithms for general graphs still remains open [20].…”