An Improved Approximation Algorithm for the Minimum Cost Subset k-Connected Subgraph Problem

Laekhanukit, Bundit

doi:10.1007/s00453-014-9869-5

Cited by 11 publications

(11 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The rooted k-connectivity problem is a fundamental network design problem with vertex-connectivity requirements, and it lies at the bottom of the complexity hierarchy of the vertex-connectivity problems. In particular, the undirected rooted k-connectivity problem was shown to be a special case of the subset k-connectivity problem [20] and is clearly a special case of the vertex-connectivity survivable network design problem. It can be seen that the same relationships also apply for the case of directed graphs.…”

Section: Introductionmentioning

confidence: 99%

Parameters of Two-Prover-One-Round Game and The Hardness of Connectivity Problems

Laekhanukit

2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

Self Cite

View full text Add to dashboard Cite

Optimizing parameters of Two-Prover-One-Round Game (2P1R) is an important task in PCPs literature as it would imply a smaller PCP with the same or stronger soundness. While this is a basic question in PCPs community, the connection between the parameters of PCPs and hardness of approximations is sometimes obscure to approximation algorithm community. In this paper, we investigate the connection between the parameters of 2P1R and the hardness of approximating the class of so-called connectivity problems, which includes as subclasses the survivable network design and (multi)cut problems. Based on recent development on 2P1R by Chan (STOC 2013) and several techniques in PCPs literature, we improve hardness results of some connectivity problems that are in the form k σ , for some (very) small constant σ > 0, to hardness results of the form k c for some explicit constant c, where k is a connectivity parameter. In addition, we show how to convert these hardness into hardness results of the form D c , where D is the number of demand pairs (or the number of terminals). Our results are as follows.

show abstract

Section: Introductionmentioning

confidence: 99%

Parameters of Two-Prover-One-Round Game and The Hardness of Connectivity Problems

Laekhanukit

2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

Self Cite

View full text Add to dashboard Cite

show abstract

“…Approximation algorithms for 2-DST can be used to approximate with the same asymptotic approximation factor the more general problem, namely 2-DSS, described in [11,35,41] (see Appendix A for more details). Definition 3.…”

Section: Our Results and Techniquesmentioning

confidence: 99%

Surviving in Directed Graphs: A Polylogarithmic Approximation for Two-Connected Directed Steiner Tree

Grandoni¹,

Laekhanukit²

2016

Preprint

Self Cite

View full text Add to dashboard Cite

Real-word networks are often prone to failures. A reliable network needs to cope with this situation and must provide a backup communication channel. This motivates the study of survivable network design, which has been a focus of research for a few decades. To date, survivable network design problems on undirected graphs are well-understood. For example, there is a 2 approximation in the case of edge failures [Jain, FOCS'98/Combinatorica'01]. The problems on directed graphs, in contrast, have seen very little progress. Most techniques for the undirected case like primal-dual and iterative rounding methods do not seem to extend to the directed case. Almost no non-trivial approximation algorithm is known even for a simple case where we wish to design a network that tolerates a single failure.In this paper, we study a survivable network design problem on directed graphs, 2-Connected Directed Steiner Tree (2-DST): given an n-vertex weighted directed graph, a root r, and a set of h terminals S, find a min-cost subgraph H that has two edge/vertex disjoint paths from r to any t ∈ S. 2-DST is a natural generalization of the classical Directed Steiner Tree problem (DST), where we have an additional requirement that the network must tolerate one failure. No non-trivial approximation is known for 2-DST. This was left as an open problem by Feldman et al., [SODA'09; JCSS] and has then been studied by Cheriyan et al. [SODA'12; TALG] and Laekhanukit [SODA'14]. However, no positive result was known except for the special case of a D-shallow instance [Laekhanukit, ICALP'16].We present an O(D 3 log D • h 2/D • log n) approximation algorithm for 2-DST that runs in time O(n O(D) ), for any D ∈ [log 2 h]. This implies a polynomial-time O(h ε log n) approximation for any constant ε > 0, and a poly-logarithmic approximation running in quasi-polynomial time. We remark that this is essentially the best-known even for the classical DST, and the latter problem is O(log 2−ε n)-hard to approximate [Halperin and Krauthgamer, STOC'03]. As a by product, we obtain an algorithm with the same approximation guarantee for the 2-Connected Directed Steiner Subgraph problem, where the goal is to find a min-cost subgraph such that every pair of terminals are 2-edge/vertex connected.Our approximation algorithm is based on a careful combination of several techniques. In more detail, we decompose an optimal solution into two (possibly not edge disjoint) divergent trees that induces two edge disjoint paths from the root to any given terminal. These divergent trees are then embedded into a shallow tree by means of Zelikovsky's height reduction theorem. On the latter tree we solve a 2-Connected Group Steiner Tree problem and then map back this solution to the original graph. Crucially, our tree embedding is achieved via a probabilistic mapping guided by an LP: This is the main technical novelty of our approach, and might be useful for future work.

show abstract

“…For two important special cases of VC-SNDP, namely, Rooted-VC-SNDP and Subset-VC-SNDP, O (k log k )-approximations are known in the edge-weighted case [29,32] and the node-weighted case requires an additional O (log n)-factor. It would be interesting to show that this additional factor is unnecessary in planar graphs-we note that the results in [29,32] are based on the augmentation framework and hence some of our ideas may be applicable.…”

Section: Discussionmentioning

confidence: 99%

Node-weighted Network Design in Planar and Minor-closed Families of Graphs

Chekuri

Ene

Vakilian

2021

ACM Trans. Algorithms

View full text Add to dashboard Cite

We consider node-weighted survivable network design (SNDP) in planar graphs and minor-closed families of graphs. The input consists of a node-weighted undirected graph G = ( V , E ) and integer connectivity requirements r ( uv ) for each unordered pair of nodes uv . The goal is to find a minimum weighted subgraph H of G such that H contains r ( uv ) disjoint paths between u and v for each node pair uv . Three versions of the problem are edge-connectivity SNDP (EC-SNDP), element-connectivity SNDP (Elem-SNDP), and vertex-connectivity SNDP (VC-SNDP), depending on whether the paths are required to be edge, element, or vertex disjoint, respectively. Our main result is an O ( k )-approximation algorithm for EC-SNDP and Elem-SNDP when the input graph is planar or more generally if it belongs to a proper minor-closed family of graphs; here, k = max uv r ( uv ) is the maximum connectivity requirement. This improves upon the O ( k log n )-approximation known for node-weighted EC-SNDP and Elem-SNDP in general graphs [31]. We also obtain an O (1) approximation for node-weighted VC-SNDP when the connectivity requirements are in {0, 1, 2}; for higher connectivity our result for Elem-SNDP can be used in a black-box fashion to obtain a logarithmic factor improvement over currently known general graph results. Our results are inspired by, and generalize, the work of Demaine, Hajiaghayi, and Klein [13], who obtained constant factor approximations for node-weighted Steiner tree and Steiner forest problems in planar graphs and proper minor-closed families of graphs via a primal-dual algorithm.

show abstract

An Improved Approximation Algorithm for the Minimum Cost Subset k-Connected Subgraph Problem

Cited by 11 publications

References 31 publications

Parameters of Two-Prover-One-Round Game and The Hardness of Connectivity Problems

Parameters of Two-Prover-One-Round Game and The Hardness of Connectivity Problems

Surviving in Directed Graphs: A Polylogarithmic Approximation for Two-Connected Directed Steiner Tree

Node-weighted Network Design in Planar and Minor-closed Families of Graphs

Contact Info

Product

Resources

About