Approximating Minimum-Cost Connectivity Problems

Kortsarz, Guy; Nutov, Zeev

doi:10.1201/9781420010749.ch58

Cited by 74 publications

(83 citation statements)

References 49 publications

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“…Then λ S G (u, v) = min{|C| : C ⊆ E + S − {u, v}, G − C has no uv-path} . We prove the Theorem for the directed case and the statement for the undirected CA follows from the following proposition (c.f., [16]), which implies that undirected CA problems cannot be much harder to approximate than the directed ones.…”

Section: Proof Of the Theoremmentioning

confidence: 99%

“…For rooted {0, 1}-requirements (this is the Directed Steiner Tree problem) [2] gave an O(n ε /ε 3 )-approximation algorithm for any constant ε > 0. See also surveys in [15,16] on various GSN problems.…”

Section: Related Workmentioning

confidence: 99%

“…For work on other types of GSN costs see detailed surveys in [15,16] on known upper and lower bounds with respect to approximation.…”

Section: Related Workmentioning

confidence: 99%

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Tight Approximation Algorithm for Connectivity Augmentation Problems

Kortsarz

Nutov

2006

Automata, Languages and Programming

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The S-connectivity λ S G (u, v) of (u, v) in a graph 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 G0 = (V, E0), S ⊆ V , and requirements r(u, v) on V ×V , find a minimum size set F of new edges (any edge is allowed) so thatExtensively studied particular choices of S are the edge-CA (when S = ∅) and the node-CA (when S = V ). A. Frank gave a polynomial algorithm for undirected edge-CA and observed that the directed case even with rooted {0, 1}-requirements is at least as hard as the Set-Cover problem (in rooted requirements there is s ∈ V − S so that if r(u, v) > 0 then: u = s for directed graphs, and u = s or v = s for undirected graphs). Both directed and undirected node-CA have approximation threshold Ω(2 log 1−ε n ). The only polylogarithmic approximation ratio known for CA was for rooted requirements -O(log n · log rmax) = O(log 2 n), where rmax = maxu,v∈V r (u, v). No nontrivial approximation algorithms were known for directed CA even for r(u, v) ∈ {0, 1}, nor for undirected CA with S arbitrary. We give an approximation algorithm for the general case that matches the known approximation thresholds. For both directed and undirected CA with arbitrary requirements our approximation ratio is: O(log n) for S = V arbitrary, and O(rmax · log n) for S = V .

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Section: Proof Of the Theoremmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

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Tight Approximation Algorithm for Connectivity Augmentation Problems

Kortsarz

Nutov

2006

Automata, Languages and Programming

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“…Most of the corresponding optimization problems are NP-hard. Thus, there are numerous results on polynomialtime approximability [26]. By way of contrast, the study of the parameterized complexity of these problems is much less developed (refer to [3], [12], [21] for fixed-parameter tractability and to [11] for parameterized hardness results concerning the undirected case).…”

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confidence: 99%

“…Introduction. Steiner-type problems lie at the heart of network design and connectivity problems [26] (see [30] for a broad account on Steiner tree problems). Roughly speaking, the task in these problems is to find in a given weighted graph a low-cost subgraph that satisfies prescribed connectivity requirements.…”

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confidence: 99%

Parameterized Complexity of Arc-Weighted Directed Steiner Problems

Guo

Niedermeier

Suchý

2009

Algorithms and Computation

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Abstract. We start a systematic parameterized computational complexity study of three NP-hard network design problems on arc-weighted directed graphs: directed Steiner tree, strongly connected Steiner subgraph, and directed Steiner network. We investigate their parameterized complexities with respect to the three parameterizations: "number of terminals," "an upper bound on the size of the connecting network," and the combination of these two. We achieve several parameterized hardness results as well as some fixed-parameter tractability results, in this way extending previous results of Feldman and Ruhl [SIAM J. Comput., 36 (2006) 1. Introduction. Steiner-type problems lie at the heart of network design and connectivity problems [26] (see [30] for a broad account on Steiner tree problems). Roughly speaking, the task in these problems is to find in a given weighted graph a low-cost subgraph that satisfies prescribed connectivity requirements. Most of the corresponding optimization problems are NP-hard. Thus, there are numerous results on polynomialtime approximability [26]. By way of contrast, the study of the parameterized complexity of these problems is much less developed (refer to [3], [12], [21] for fixed-parameter tractability and to [11] for parameterized hardness results concerning the undirected case). Our work contributes new algorithmic and computational hardness results concerning the parameterized complexity of three fundamental NP-hard Steiner problems in arc-weighted directed graphs.

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Approximating Minimum Power Covers of Intersecting Families and Directed Connectivity Problems

Nutov

2006

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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Given a (directed) graph with costs on the edges, the power of a node is the maximum cost of an edge leaving it, and the power of the graph is the sum of the powers of its nodes. Motivated by applications for wireless networks, we consider fundamental directed connectivity network design problems under the power minimization criteria. Let G = (V, E) be a graph with edge-costs {c(e) : e ∈ E} and let k be an integer. We consider finding a minimum power subgraph G of G that satisfies some prescribed property. The Min-Power k-Outconnected Subgraph (MPk-OS) problem requires that G contains k pairwise internally-disjoint rv-paths for all v ∈ V − r, for a given "root" r ∈ V . The Min-Power k-Connected Subgraph (MPk-CS) problem requires that G contains k pairwise internally-disjoint paths between every pair of its nodes. In the edge connectivity variant the paths are required to be only edge disjoint. For k = 1 all these problems are at least as hard as the Set-Cover problem and thus have an Ω(ln |V |) approximation threshold. For k = Θ(n) the edge connectivity variants are unlikely to admit a polylogarithmic approximation algorithm [23]. We give an O(k ln |V |)-approximation algorithms for these four problems. Our algorithms are based on a much more general O(ln |V |)-approximation algorithm for the problem of finding a min-power directed (edge-)cover of an intersecting set-family; a set-family F is intersecting if X ∩ Y, X ∪ Y ∈ F for any intersecting X, Y ∈ F, and an edge set F covers F if for every X ∈ F there is an edge in F entering X.

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Approximating Minimum-Cost Connectivity Problems

Cited by 74 publications

References 49 publications

Tight Approximation Algorithm for Connectivity Augmentation Problems

Tight Approximation Algorithm for Connectivity Augmentation Problems

Parameterized Complexity of Arc-Weighted Directed Steiner Problems

Approximating Minimum Power Covers of Intersecting Families and Directed Connectivity Problems

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