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Approximating k-forest with resource augmentation: A primal-dual approach

Abstract : In this paper, we study the k-forest problem in the model of resource augmentation. In the k-forest problem, given an edge-weighted graph G(V, E), a parameter k, and a set of m demand pairs ⊆ V× V, the objective is to construct a minimum-cost subgraph that connects at least k demands. The problem is hard to approximate—the best-known approximation ratio is O(min{√n,√k}). Furthermore, k-forest is as hard to approximate as the notoriously-hard densest k-subgraph problem. While the k-forest problem is hard to approximate in the worst-case, we show that with the use of resource augmentation, we can efficiently approximate it up to a constant factor. First, we restate the problem in terms of the number of demands that are not connected. In particular, the objective of the k-forest problem can be viewed as to remove at most m- k demands and find a minimum-cost subgraph that connects the remaining demands. We use this perspective of the problem to explain the performance of our algorithm (in terms of the augmentation) in a more intuitive way. Specifically, we present a polynomial-time algorithm for the k-forest problem that, for every ε> 0, removes at most m- k demands and has cost no more than O(1/ε2) times the cost of an optimal algorithm that removes at most (1 - ε) (m- k) demands.
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https://hal.archives-ouvertes.fr/hal-01689429
Contributor : Frédéric Davesne <>
Submitted on : Monday, January 22, 2018 - 10:07:46 AM
Last modification on : Saturday, May 1, 2021 - 3:48:50 AM

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Eric Angel, Nguyen Kim Thang, Shikha Singh. Approximating k-forest with resource augmentation: A primal-dual approach. 11th International Conference on Combinatorial Optimization and Applications (COCOA 2017), Dec 2017, Shangai, China. pp.333--347, ⟨10.1007/978-3-319-71147-8_23⟩. ⟨hal-01689429⟩

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