On the disjoint paths problem

Nguyen, Thinh D.

doi:10.1016/j.orl.2006.02.001

Cited by 11 publications

(9 citation statements)

References 4 publications

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“…We prove Theorem 1.1 in Section 3, after setting up some notation and proving a lemma on single source flows in Section 2. We remark that Thanh Nguyen [16] has independently obtained an approximation ratio of O( √ n) for DAGs, and subsequent to our work, obtained an alternative O( √ n) approximation in undirected graphs.…”

Section: Corollary 12 the Integrality Gap Of The Relaxation Based Omentioning

confidence: 52%

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Chekuri¹,

Khanna²,

Shepherd³

2006

Theory of Comput.

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We consider the maximization version of the edge-disjoint path problem (EDP). In undirected graphs and directed acyclic graphs, we obtain an O(√ n) upper bound on the approximation ratio where n is the number of nodes in the graph. We show this by establishing the upper bound on the integrality gap of the natural relaxation based on multicommodity flows. Our upper bound matches within a constant factor a lower bound of Ω(√ n) that is known for both undirected and directed acyclic graphs. The best previous upper bounds on the integrality gaps were O(min{n 2/3 , √ m}) for undirected graphs and O(min{ √ n log n, √ m}) for directed acyclic graphs; here m is the number of edges in the graph. These bounds are also the best known approximation ratios for these problems. Our bound also extends to the unsplittable flow problem (UFP) when the maximum demand is at most the minimum capacity.

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Section: Corollary 12 the Integrality Gap Of The Relaxation Based Omentioning

confidence: 52%

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Chekuri¹,

Khanna²,

Shepherd³

2006

Theory of Comput.

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“…Namely, all of the lower bound instances have a linear number of edges m = O(n). Therefore, it is possible that there exist upper bounds dependent on √ n. Indeed, for the special case of MEDP in undirected graphs and directed acyclic graphs O( √ n)-approximations have been developed [9,28].…”

Section: Unsplittable Flow With Arbitrary Demandsmentioning

confidence: 99%

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Shepherd¹,

Vetta²

2017

Theory of Comput.

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“…Instances with only root-using demand paths can be solved using a matching approach, see e.g. the work of Nguyen [21]. To solve all-ror instances in general we use bidirected flows, which were introduced by Edmonds and Johnson [9].…”

Section: Definitionmentioning

confidence: 99%

Max-Weight Integral Multicommodity Flow in Spiders and High-Capacity Trees

Könemann

Parekh

Pritchard

2009

Approximation and Online Algorithms

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Abstract. We consider the max-weight integral multicommodity flow problem in trees. In this problem we are given an edge-capacitated tree and weighted pairs of terminals, and the objective is to find a max-weight integral flow between terminal pairs subject to the capacities. This problem was shown to be APX-hard by Garg, Vazirani and Yannakakis [Algorithmica, 1997], and a 4-approximation was given by Chekuri, Mydlarz and Shepherd [ACM Trans. Alg., 2007]. Some special cases are known to be exactly solvable in polynomial time, including when the graph is a path or a star. First, when every edge has capacity at least µ ≥ 2, we use iterated LP relaxation to obtain an improved approximation ratio of min{3, 1 + 4/µ + 6/(µ 2 − µ)}. We show this ratio bounds the integrality gap of the natural LP relaxation. A complementary hardness result yields a 1 + Θ(1/µ) threshold of approximability (if P = NP). Second, we extend the range of instances for which exact solutions can be found efficiently. When the tree is a spider (i.e. if only one vertex has degree greater than 2) we give a polynomial-time algorithm to find an optimal solution, as well as a polyhedral description of the convex hull of all integral feasible solutions.

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On the disjoint paths problem

Cited by 11 publications

References 4 publications

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Max-Weight Integral Multicommodity Flow in Spiders and High-Capacity Trees

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