Increasing digraph arc-connectivity by arc addition, reversal and complement

Arkin, Esther M.; Hassin, Refael; Shahar, Shimon

doi:10.1016/s0166-218x(01)00319-5

Cited by 3 publications

(7 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Formally, let N be the set of natural numbers, and let W ⊆ N ∪ f0; ∞g. If V is a set of vertices, w∶V × V → W is a weight function, 1 and A ⊆ V × V is a set of arcs; then we define wðAÞ ≔ P a∈A wðaÞ. We study the following: Directed Steiner Tree (DST): Instance: A set of vertices V , a weight function w∶V × V → W , a set T ⊆ V of terminals (l ≔ jT j), a root s ∈ V , and a weight bound p ∈ N. Question: Is there a set of arcs A ⊆ V × V of weight wðAÞ ≤ p such that in the digraph D ≔ ðV ; AÞ for every t ∈ T there is a directed path from s to t?…”

mentioning

confidence: 99%

“…• the weight p of the solution divided by the minimum arc weight min W (again excluding 0), giving the parameter p∕ min W ; 2 and • the combined parameter ðl; p∕ min W Þ. Note that a parameterized hardness result with respect to the combined parameter clearly means hardness results for each single parameter and, on the contrary, a 1 Observe that in this way we implicitly deal with complete digraphs in the sense that only arc weights are specified. 2 This parameter naturally reflects the number of arcs in the spanning subgraph by providing an upper bound on the number of (nonzero) arcs.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Parameterized Complexity of Arc-Weighted Directed Steiner Problems

Guo¹,

Niedermeier²,

Suchý³

2011

SIAM J. Discrete Math.

View full text Add to dashboard Cite

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 sub-graph, 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), pp. 543-561]. 1. Introduction. Steiner-type problems lie at the heart of network design and con-nectivity 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 polynomial-time 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.

show abstract

mentioning

confidence: 99%

mentioning

confidence: 99%

Parameterized Complexity of Arc-Weighted Directed Steiner Problems

Guo¹,

Niedermeier²,

Suchý³

2011

SIAM J. Discrete Math.

View full text Add to dashboard Cite

show abstract

“…Specifically, the parameterized complexity of (0-)SCSS and DSN is unsettled for some small values of the arc-weight ratio. Given the vast literature on polynomial-time approximation algorithms, one may encounter many more questions to study concerning the parameterized complexity of network design problems in general-for instance, the connectivity augmentation problems with arc reversal and complement operations [1]. Moreover, it would also be interesting to investigate whether some restrictions on the graph structure, such as planarity, could lead to fixed-parameter tractability results of Steiner-type problems (see Bateni, Hajiaghayi, and Marx [2] for some approximation results).…”

Section: Algorithmic Resultsmentioning

confidence: 99%

“…Let ðI 1 ; l; dÞ; : : : ; ðI r ; l; dÞ be a set of DST instances where, for 1 ≤ i ≤ r, I i consists of a vertex set V i , a weight function w i ∶V i × V i → W i , a set T i ⊆ V i of terminals, a root s i ∈ V i , and a weight bound p i ∈ N. Moreover, l ¼ jT 1 …”

Section: Preliminariesmentioning

confidence: 99%

“…Note that a parameterized hardness result with respect to the combined parameter clearly means hardness results for each single parameter and, on the contrary, a 1 Observe that in this way we implicitly deal with complete digraphs in the sense that only arc weights are specified. 2 This parameter naturally reflects the number of arcs in the spanning subgraph by providing an upper bound on the number of (nonzero) arcs.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Parameterized Complexity of Arc-Weighted Directed Steiner Problems

Guo

Niedermeier

Suchý

2009

Algorithms and Computation

View full text Add to dashboard Cite

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.

show abstract

Making Bidirected Graphs Strongly Connected

Matsuoka

Sato

2019

Algorithmica

View full text Add to dashboard Cite

We consider problems to make a given bidirected graph strongly connected with minimum cardinality of additional signs or additional arcs. For the former problem, we show the minimum number of additional signs and give a linear-time algorithm for finding an optimal solution. For the latter problem, we give a linear-time algorithm for finding a feasible solution whose size is equal to the obvious lower bound or more than that by one.

show abstract

Increasing digraph arc-connectivity by arc addition, reversal and complement

Cited by 3 publications

References 10 publications

Parameterized Complexity of Arc-Weighted Directed Steiner Problems

Parameterized Complexity of Arc-Weighted Directed Steiner Problems

Parameterized Complexity of Arc-Weighted Directed Steiner Problems

Making Bidirected Graphs Strongly Connected

Contact Info

Product

Resources

About