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

Grandoni, Fabrizio; Mömke, Tobias; Wiese, Andreas; Zhou, Hang

doi:10.1137/1.9781611974782.159

Cited by 7 publications

(7 citation statements)

References 23 publications

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“…Extending this approach, Grandoni et al [22] present PTASs for the special cases that there is an edge that is used by all tasks, that there are no two tasks whose paths are contained in each other, that one can select an arbitrary number of copies of each task, and that the profit of each task is proportional to the product of its demand and the length of its path, i.e., its area in a geometric sense. Very recently, Grandoni et al [23] found a polynomial time (31/16 + )-approximation algorithm for UFP.…”

Section: Follow-up Resultsmentioning

confidence: 99%

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

Anagnostopoulos

Grandoni

Leonardi

et al. 2018

ACM Trans. Algorithms

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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.

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Section: Follow-up Resultsmentioning

confidence: 99%

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

Anagnostopoulos

Grandoni

Leonardi

et al. 2018

ACM Trans. Algorithms

Self Cite

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“…This may allow tractable formulations with larger values of t. A related idea is to consider the intersection of the integer hulls of sub-instances induced by keeping a subset of the requests (instead of keeping a subset of the edges). For example, to restrict to the set of requests which pass through at least one of some set of t edges, as such instances are known to be easier to approximate [GMWZ17]. Lastly, the question of whether P k c rank has constant integrality gap for general ANF-Tree instances has so far eluded us; it remains a very interesting question.…”

Section: Discussionmentioning

confidence: 99%

A Knapsack Intersection Hierarchy Applied to All-or-Nothing Flow in Trees

Jozefiak¹,

Shepherd²,

Weninger³

2022

Preprint

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We introduce a natural knapsack intersection hierarchy for strengthening linear programming relaxations of packing integer programs, i.e., max{w T x : x ∈ P ∩ {0, 1} n } where P = {x ∈ [0, 1] n : Ax ≤ b} and A, b, w ≥ 0. The t th level P t corresponds to adding cuts associated with the integer hull of the intersection of any t knapsack constraints (rows of the constraint matrix). This model captures the maximum possible strength of "t-row cuts", an approach often used by solvers for small t. If A is m × n, then P m is the integer hull of P and P 1 corresponds to adding cuts for each associated single-row knapsack problem. Thus, even separating over P 1 is NP-hard. However, for fixed t and any ǫ > 0, results of Pritchard imply there is a polytime (1 + ǫ)-approximation for P t . We then investigate the hierarchy's strength in the context of the well-studied all-or-nothing flow problem in trees (also called unsplittable flow on trees). For this problem, we show that the integrality gap of P t is O(n/t) and give examples where the gap is Ω(n/t). We then examine the stronger formulation P rank where all rank constraints are added. For P t rank , our best lower bound drops to Ω(1/c) at level t = n c for any c > 0. Moreover, on a well-known class of "bad instances" due to Friggstad and Gao, we show that we can achieve this gap; hence a constant integrality gap for these instances is obtained at level n c .

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“…Once the stairway is fixed we can identify and distinguish between locally large and small items. This is the main difference between our approach here and the techniques used for UFP and related problems [1,5,10], or the techniques used in other works on IIK [7,23].…”

Section: Iik Mkp and Ufpmentioning

confidence: 99%

A PTAS for the Time-Invariant Incremental Knapsack Problem

Faenza

Malinović

2018

Lecture Notes in Computer Science

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The Time-Invariant Incremental Knapsack problem (IIK) is a generalization of Maximum Knapsack to a discrete multi-period setting. At each time, capacity increases and items can be added, but not removed from the knapsack. The goal is to maximize the sum of profits over all times. IIK models various applications including specific financial markets and governmental decision processes. IIK is strongly NP-hard [7] and there has been work [7,8,11,21,23] on giving approximation algorithms for some special cases. In this paper, we settle the complexity of IIK by designing a PTAS based on rounding a disjuncive formulation, and provide several extensions of the technique.

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

Cited by 7 publications

References 23 publications

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

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

A Knapsack Intersection Hierarchy Applied to All-or-Nothing Flow in Trees

A PTAS for the Time-Invariant Incremental Knapsack Problem

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