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Chekuri, Chandra; Khanna, Sanjeev; Shepherd, F. Bruce

doi:10.4086/toc.2006.v002a007

Cited by 71 publications

(13 citation statements)

References 15 publications

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“…This was later generalized to ufp by Srinivasan [19] and Baveja and Srinivasan [6] under no-bottleneck assumption: max i d i ≤ min e c e . More recently, Chekuri et al [9] improved these results to O( |V |)-approximation. On the other hand, Guruswami et al [14] proved that edp on directed graphs is NP-hard to approximate within a factor of Ω(|E| 1 2 −ǫ ) for any constant ǫ > 0.…”

Section: Introductionmentioning

confidence: 94%

A logarithmic approximation for unsplittable flow on line graphs

Bansal

Friggstad

Khandekar

et al. 2014

ACM Trans. Algorithms

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We consider the unsplittable flow problem on a line. In this problem, we are given a set of n tasks, each specified by a start time si, an end time ti, a demand di > 0, and a profit pi > 0. A task, if accepted, requires di units of "bandwidth" from time si to ti and accrues a profit of pi. For every time t, we are also specified the available bandwidth ct, and the goal is to find a subset of tasks with maximum profit subject to the bandwidth constraints.In this paper, we present the first polynomial-time O(log n)-approximation algorithm for this problem. No polynomial-time o(n)-approximation was known prior to this work. Previous results for this problem were known only in more restrictive settings, in particular, either if the given instance satisfies the so-called "no-bottleneck" assumption: maxi di ≤ mint ct, or else if the ratio of the maximum to the minimum demands and ratio of the maximum to the minimum capacities are polynomially (or quasi-polynomially) bounded in n. Our result, on the other hand, does not require any of these assumptions.Our algorithm is based on a combination of dynamic programming and rounding a natural linear programming relaxation for the problem. While there is an Ω(n) integrality gap known for this LP relaxation, our key idea is to exploit certain structural properties of the problem to show that instances that are bad for the LP can in fact be handled using dynamic programming.

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Section: Introductionmentioning

confidence: 94%

A logarithmic approximation for unsplittable flow on line graphs

Bansal

Friggstad

Khandekar

et al. 2014

ACM Trans. Algorithms

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“…Disjoint paths problems have also been studied intensively in the area of approximation algorithms, both on directed and undirected graphs (see, e.g., [9,18,2,5,8,4,6,10,7]). The goal is, given an input graph G and demands (s 1 , t 1 ), .…”

Section: Introductionmentioning

confidence: 99%

Routing with congestion in acyclic digraphs

Amiri

Kreutzer

Marx

et al. 2019

Information Processing Letters

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We study the version of the k-disjoint paths problem where k demand pairs (s 1 , t 1),. .. , (s k , t k) are specified in the input and the paths in the solution are allowed to intersect, but such that no vertex is on more than c paths. We show that on directed acyclic graphs the problem is solvable in time n O(d) if we allow congestion k − d for k paths. Furthermore, we show that, under a suitable complexity theoretic assumption, the problem cannot be solved in time f (k)n o(d/ log d) for any computable function f .

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“…In this paper, we consider the optimization problems MaxEDP and MaxNDP when the number of pairs is part of the input. In this setting, the best approximation ratio for MaxEDP is achieved by an -approximation algorithm [11, 35], that is, by an algorithm that routes pairs, where is the number of pairs in an optimum routing and n is the number of nodes. However, the best known lower bound for undirected graphs is only , assuming [19].…”

Section: Introductionmentioning

confidence: 99%

“…The approximability of MaxEDP/MaxNDP is currently not well understood; the best known lower bound is , assuming . This constitutes a significant gap to the best known approximation upper bound of due to Chekuri et al (Theory Comput 2:137–146, 2006), and closing this gap is currently one of the big open problems in approximation algorithms. In their seminal paper, Raghavan and Thompson (Combinatorica 7(4):365–374, 1987) introduce the technique of randomized rounding for LPs; their technique gives an -approximation when edges (or nodes) may be used by paths.…”

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confidence: 99%

New algorithms for maximum disjoint paths based on tree-likeness

2017

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We study the classical NP-hard problems of finding maximum-size subsets from given sets of k terminal pairs that can be routed via edge-disjoint paths (MaxEDP) or node-disjoint paths (MaxNDP) in a given graph. The approximability of MaxEDP/MaxNDP is currently not well understood; the best known lower bound is 2 Ω( √ log n) , assuming NP DTIME (n O(log n) (or nodes) may be used by O (log n/ log log n) paths. In this paper, we strengthen the fundamental results above. We provide new bounds formulated in terms of the feedback vertex set number r of a graph, which measures its vertex deletion distance to a forest. In particular, we obtain the following results:-For MaxEDP, we give an O( √ r log(kr))-approximation algorithm. Up to a logarithmic factor, our result strengthens the best known ratio O( √ n) due to Chekuri et al., as r ≤ n. -Further, we show how to route Ω(OPT * ) pairs with congestion bounded by O(log(kr)/ log log(kr)), strengthening the bound obtained by the classic approach of Raghavan and Thompson. -For MaxNDP, we give an algorithm that gives the optimal answer in time

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Cited by 71 publications

References 15 publications

A logarithmic approximation for unsplittable flow on line graphs

A logarithmic approximation for unsplittable flow on line graphs

Routing with congestion in acyclic digraphs

New algorithms for maximum disjoint paths based on tree-likeness

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