A New Approach to Splitting-Off

Bernáth, Attila; Király, Tamás

doi:10.1007/978-3-540-68891-4_28

Cited by 5 publications

(2 citation statements)

References 10 publications

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“…We note that recently Bernáth and Király [Bernáth and Király 2008] showed several extensions and other interesting applications of the new approach developed in this paper, including simplified proofs of many classic results, as well as some new interesting results. It is not clear however that all possible applications of our approach are exhausted.…”

Section: Set-function Edge-covermentioning

confidence: 73%

“…Our original proof for the fact that p is 0, 1-valued in the preliminary version [Nutov 2005] was somewhat long and complicated, and recently [Bernáth and Király 2008] found a much simpler and more constructive proof. We see no point in presenting our original proof, and for this part present a proof along the proof line of [Bernáth and Király 2008].…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

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Approximating connectivity augmentation problems

Nutov

2009

ACM Trans. Algorithms

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Let G = (V, E) be an undirected graph and let S ⊆ V . The S-connectivity λ S G (u, v) of a node pair (u, v) in G is the maximum number of uv-paths that no two of them have an edge or a node in S − {u, v} in common. The corresponding Connectivity Augmentation (CA) problem is: given a graph G = (V, E), a node subset S ⊆ V , and a nonnegative integer requirement function r(u, v) on V ×V , add a minimum size set F of new edges to G so that λ S G+F (u, v) ≥ r(u, v) for all (u, v) ∈ V ×V . Three extensively studied particular cases are: the Edge-CA (S = ∅), the Node-CA (S = V ), and the Element-CA (r(u, v) = 0 whenever u ∈ S or v ∈ S). A polynomial algorithm for Edge-CA was developed by Frank. In this paper we consider the Element-CA and the Node-CA, that are NP-hard even for r(u, v) ∈ {0, 2}. The best known ratios for these problems were: 2 for Element-CA and O(rmax · ln n) for Node-CA, where rmax = max u,v∈V r(u, v) and n = |V |. Our main result is a 7/4-approximation algorithm for the Element-CA, improving the previously best known 2-approximation. For Element-CA with r(u, v) ∈ {0, 1, 2} we give a 3/2-approximation algorithm. These approximation ratios are based on a new splitting-off theorem, which implies an improved lower bound on the number of edges needed to cover a skew-supermodular set function. For Node-CA we establish the following approximation threshold: Node-CA with r(u, v) ∈ {0, k} cannot be approximated within O(2 log 1−ε n ) for any fixed ε > 0, unless NP ⊆ DTIME(n polylog(n) ).

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Section: Set-function Edge-covermentioning

confidence: 73%

Section: Proof Of Theorem 13mentioning

confidence: 99%

Approximating connectivity augmentation problems

Nutov

2009

ACM Trans. Algorithms

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Efficient Edge Splitting-Off Algorithms Maintaining All-Pairs Edge-Connectivities

Lau

Yung

2010

Integer Programming and Combinatorial Optimization

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Abstract. In this paper we present new edge splitting-off results maintaining all-pairs edge-connectivities of a graph. We first give an alternate proof of Mader's theorem, and use it to obtain a deterministic O(rmax 2 · n 2 )-time complete edge splitting-off algorithm for unweighted graphs, where rmax denotes the maximum edge-connectivity requirement. This improves upon the best known algorithm by Gabow by a factor of Ω(n). We then prove a new structural property, and use it to further speedup the algorithm to obtain a randomizedÕ(m + rmax 3 · n)-time algorithm. These edge splitting-off algorithms can be used directly to speedup various graph algorithms.

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Augmenting Edge-Connectivity between Vertex Subsets

Ishii

Makino

2012

Algorithmica

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Given a graph G = (V, E) and a requirement function r : W 1 × W 2 → R + for two families W 1 , W 2 ⊆ 2 V − {∅}, we consider the problem (called area-toarea edge-connectivity augmentation problem) of augmenting G by a smallest number of new edges so that the resulting graphĜ satisfies, where δ G (X) denotes the degree of a vertex set X in G. This problem can be regarded as a natural generalization of the global, local, and node-to-area edge-connectivity augmentation problems.In this paper, we show that there exists a constant c such that the problem is inapproximable within a ratio of c log α(W 1 , W 2 ), unless P=NP, even restricted to the directed global node-to-area edge-connectivity augmentation or undirected local node-to-area edge-connectivity augmentation, where α(W 1 , W 2 ) denotes the number of pairs W 1 ∈ W 1 and W 2 ∈ W 2 with r(W 1 , W 2 ) > 0. We also provide an O(log α(W 1 , W 2 ))-approximation algorithm for the area-to-area edge-connectivity augmentation problem. This together with the negative result implies that the problem is Θ(log α(W 1 , W 2 ))-approximable, unless P=NP, which solves open problems for nodeto-area edge-connectivity augmentation , Ishii and Hagiwara 2006, Miwa and Ito 2004.Furthermore, we characterize the node-toarea and area-to-area edge-connectivity augmentation problems as the augmentation problems with modulotone and extended modulotone functions.

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A New Approach to Splitting-Off

Cited by 5 publications

References 10 publications

Approximating connectivity augmentation problems

Approximating connectivity augmentation problems

Efficient Edge Splitting-Off Algorithms Maintaining All-Pairs Edge-Connectivities

Augmenting Edge-Connectivity between Vertex Subsets

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