A logarithmic approximation for unsplittable flow on line graphs

Bansal, Nikhil; Friggstad, Zachary; Khandekar, Rohit; Salavatipour, Mohammad R.

doi:10.1145/2532645

Cited by 19 publications

(27 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The best known polynomial-time approximation algorithm for UFP prior to this work achieved an approximation factor of 7 + [12]. This result improves on the previously best known polynomial time (log )-approximation algorithm designed by Bansal et al [7]. Bansal et al [6] present a QPTAS for UFP assuming a quasi-polynomial bound on capacities and demands of the input instance.…”

Section: Related Worksupporting

confidence: 58%

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

Anagnostopoulos

Grandoni

Leonardi

et al. 2018

ACM Trans. Algorithms

View full text Add to dashboard Cite

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.

show abstract

Section: Related Worksupporting

confidence: 58%

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

Anagnostopoulos

Grandoni

Leonardi

et al. 2018

ACM Trans. Algorithms

View full text Add to dashboard Cite

show abstract

“…Also, there is a QPTAS for quasi-polynomially bounded input data [1]. For UFP there is a long line of work on the case of uniform edge capacities [29,6,13], the no-bottleneck-assumption [14,16], and the general case [3,4,15,11,2] which culminated in a QPTAS [3,9], PTASs for several special cases [20,9], a (2 + ε)approximation [2], which was improved to a (5/3 + ε)-approximation [21].…”

Section: Other Related Workmentioning

confidence: 99%

“…Similarly, we can combine all corresponding sets of relatively large tasks to one boxable solution OPT (3) := OPT L,↓ ∪ OPT S,top,S ∪ OPT S,bottom,S ∪ OPT S,no−cross,S . Also, we have that OPT (4) := OPT L,↓ ∪ OPT S,sw,S is a boxable solutions. Finally, we have that OPT (5) := OPT L,↓ ∪ OPT S,stair forms a stairsolution.…”

Section: Structural Lemma Arbitrary Capacitiesmentioning

confidence: 99%

“…Hence, (T , h ) or (T , h ) satisfies the claim of the lemma. Formally, our candidate solutions are OPT (1) := OPT S , OPT (2) := OPT L ∪ OPT S,bottom,S , OPT (3) := OPT L ∪ OPT S,bottom,L , OPT (4) := OPT L ∪ OPT S,top,S OPT (5) := OPT L ∪ OPT S,top,L , OPT (6) := T box , OPT (7) := T box , OPT (8) := OPT L ∪ OPT S,cross , OPT (7) := OPT L,top ∪ OPT L,bottom ∪ OPT S,mid,top,S , OPT (8) := OPT L,top ∪ OPT L,bottom ∪ OPT S,mid,top,L , OPT (9) := OPT L,top ∪ OPT L,bottom ∪ OPT S,mid,bottom,S , OPT (10) := OPT L,top ∪ OPT L,bottom ∪ OPT S,mid,bottom,L , OPT (11) := T sw , and OPT (12) := T sw .…”

Section: Types Of Points In the Middlementioning

confidence: 99%

“…We have that OPT (1) , OPT (9) , and OPT (10) are pile boxable solutions, OPT (2) , OPT (4) , OPT (7) , and OPT (9) are laminar boxable solutions, OPT (3) , OPT (5) , OPT (8) , and OPT (10) are δ-jammed solutions, and finally OPT (7) and OPT (8) are O δ (1)-boxable solutions.…”

Section: Types Of Points In the Middlementioning

confidence: 99%

See 2 more Smart Citations

Breaking the Barrier of 2 for the Storage Allocation Problem

Mömke,

Wiese

2019

Preprint

View full text Add to dashboard Cite

Packing problems are an important class of optimization problems. The probably most well-known problem if this type is knapsack and many generalizations of it have been studied in the literature like Two-dimensional Geometric Knapsack (2DKP) and Unsplittable Flow on a Path (UFP). For the latter two problems, recently the first polynomial time approximation algorithms with better approximation ratios than 2 were presented [Gálvez et al., FOCS 2017][Grandoni et al., STOC 2018. In this paper we break the barrier of 2 for the Storage Allocation Problem (SAP) which is a natural intermediate problem between 2DKP and UFP. We are given a path with capacitated edges and a set of tasks where each task has a start vertex, an end vertex, a size, and a profit. We seek to select the most profitable set of tasks that we can draw as non-overlapping rectangles underneath the capacity profile of the edges where the height of each rectangle equals the size of the corresponding task. This problem is motivated by settings of allocating resources like memory, bandwidth, etc. where each request needs a contiguous portion of the resource.The best known polynomial time approximation algorithm for SAP has an approximation ratio of 2 + ε [Mömke and Wiese, ICALP 2015] and no better quasi-polynomial time algorithm is known. We present a polynomial time (63/32 + ε) < 1.969-approximation algorithm for the case of uniform edge capacities and a quasi-polynomial time (1.997 + ε)-approximation algorithm for non-uniform quasipolynomially bounded edge capacities. Key to our results are building blocks consisting of stair-blocks, jammed tasks, and boxes that we use to construct profitable solutions and which allow us to compute solutions of these types efficiently. Finally, using our techniques we show that under slight resource augmentation we can obtain even approximation ratios of 3/2 + ε in polynomial time and 1 + ε in quasi-polynomial time, both for arbitrary edge capacities.

show abstract