Hardness of Approximation for Vertex-Connectivity Network Design Problems

Kortsarz, Guy; Krauthgamer, Robert; Lee, James R.

doi:10.1137/s0097539702416736

Cited by 112 publications

(86 citation statements)

References 19 publications

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“…First, in Theorem 3 we prove a 2 log 1−ǫ (n) -inapproximability result for Horn minimization assuming NP ⊆DTIME(n polylog(n) ) via a reduction from the Minrep problem [26]. This seems to be the first inapproximability result for this problem.…”

Section: Contributions Of This Papermentioning

confidence: 89%

“…The reduction is from the Minrep problem [26]. An instance M is given by a bipartite graph G = (A, B, E) with |E| = m, a partition of A into equal-size subsets A 1 , A 2 , .…”

Section: Inapproximabilitymentioning

confidence: 99%

“…Let s = |A| + |B|. It is shown in Kortsarz et al [26] that Minrep is 2 log 1−ε n -inapproximable under the complexity-theoretic assumption NP ⊆DTIME(n polylog(n) ).…”

Section: Inapproximabilitymentioning

confidence: 99%

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On Approximate Horn Formula Minimization

Bhattacharya

DasGupta

Mubayi

et al. 2010

Automata, Languages and Programming

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The minimization problem for Horn formulas is to find a Horn formula equivalent to a given Horn formula, using a minimum number of clauses. A 2 log 1−ǫ (n) -inapproximability result is proven, which is the first inapproximability result for this problem. We also consider several other versions of Horn minimization. The more general version which allows for the introduction of new variables is known to be too difficult as its equivalence problem is co-NP-complete. Therefore, we propose a variant called Steinerminimization, which allows for the introduction of new variables in a restricted manner. Steiner-minimization of Horn formulas is shown to be MAX-SNP-hard. In the positive direction, a o(n), namely, O(n log log n/(log n) 1/4 )-approximation algorithm is given for the Steiner-minimization of definite Horn formulas. The algorithm is based on a new result in algorithmic extremal graph theory, on partitioning bipartite graphs into complete bipartite graphs, which may be of independent interest. Inapproximability results and approximation algorithms are also given for restricted versions of Horn minimization, where only clauses present in the original formula may be used.

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Section: Contributions Of This Papermentioning

confidence: 89%

“…The reduction is from the Minrep problem [26]. An instance M is given by a bipartite graph G = (A, B, E) with |E| = m, a partition of A into equal-size subsets A 1 , A 2 , .…”

Section: Inapproximabilitymentioning

confidence: 99%

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On Approximate Horn Formula Minimization

Bhattacharya

DasGupta

Mubayi

et al. 2010

Automata, Languages and Programming

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“…Thus part (iii) immediately implies the main result of [9]: Corollary 1.3 ( [9]) Undirected {0, k}-SNDP does not admit an O(2 log 1−ε n )-approximation for any fixed ε > 0, unless NP ⊆ quasi-P.…”

Section: Introductionmentioning

confidence: 78%

“…Specifically, Dodis and Khanna [4] showed that Directed Steiner Forest is at least as hard to approximate as Label-Cover. By extending the construction of [4], Kortsarz, Krauthgamer, and Lee [9] showed a similar hardness result for Undirected {0, k}-SNDP; the same hardness is valid even for 0, 1-costs, see [13]. Even a very special case of Directed Steiner Tree -the Group Steiner Tree problem on trees, is unlikely to admit a log 2−ε n ratio for any ε > 0 [7].…”

Section: Introductionmentioning

confidence: 82%

Inapproximability of Survivable Networks

Lando

Nutov

Lecture Notes in Computer Science

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In the Survivable Network Design Problem (SNDP) one seeks to find a minimum cost subgraph that satisfies prescribed node-connectivity requirements. We give a novel approximation ratio preserving reduction from Directed SNDP to Undirected SNDP. Our reduction extends and widely generalizes as well as significantly simplifies the main results of [9]. Using it, we derive some new hardness of approximation results, as follows. We show that directed and undirected variants of SNDP and of k-Connected Subgraph are equivalent w.r.t. approximation, and that a ρ-approximation for Undirected Rooted SNDP implies a ρ-approximation for Directed Steiner Tree.

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Kernelization and complexity results for connectivity augmentation problems

Guo

Uhlmann

2009

Networks

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Connectivity augmentation problems ask for adding a set of at most k edges (called links) whose insertion makes a given graph satisfy a specified connectivity property, such as bridge-connectivity or biconnectivity. A bridge-connected (biconnected) graph is a connected graph that does not possess an edge (a vertex) whose removal results in a disconnected graph. We show that, for bridge-connectivity and biconnectivity, the respective connectivity augmentation problems admit problem kernels with O(k 2 ) vertices and links. Moreover, we study partial connectivity augmentation problems, naturally generalizing connectivity augmentation problems. Here, we do not require that, after adding the edges, the entire graph should satisfy the connectivity property, but a large subgraph. In this setting, three polynomial-time solvable connectivity augmentation problems behave differently, namely, the partial bridge-connectivity augmentation problem and the partial biconnectivity augmentation problem remain polynomial-time solvable whereas the partial strong connectivity augmentation problem becomes W[2]-hard with respect to k.

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Hardness of Approximation for Vertex-Connectivity Network Design Problems

Cited by 112 publications

References 19 publications

On Approximate Horn Formula Minimization

On Approximate Horn Formula Minimization

Inapproximability of Survivable Networks

Kernelization and complexity results for connectivity augmentation problems

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