## Abstract

In the Group Steiner Tree problem (GST), we are given a (edge or vertex)-weighted graph G = (V,E) on n vertices, together with a root vertex r and a collection of groups {S_{t}}_{iϵ[h]} : S_{t} ∪ V (G). The goal is to find a minimum-cost subgraph H that connects the root to every group. We consider a fault-tolerant variant of GST, which we call Restricted (Rooted) Group SNDP. In this setting, each group Si has a demand ki_{2} [k], k_{2} N, and we wish to find a minimum-cost subgraph H ∪ G such that, for each group Si, there is a vertex in the group that is connected to the root via ki (vertex or edge) disjoint paths. While GST admits O(log^{2} n log h) approximation, its higher connectivity variants are known to be Label-Cover hard, and for the vertex-weighted version, the hardness holds even when k = 2 (it is widely believed that there is no subpolynomial approximation for the Label-Cover problem [Bellare et al., STOC 1993]). More precisely, the problem admits no 2log^{1-n}-approximation unless NP ∪ DTIME(n^{polylog(n)}). Previously, positive results were known only for the edgeweighted version when k ≥2 [Gupta et al., SODA 2010; Khandekar et al., Theor. Comput. Sci., 2012] and for a relaxed variant where ki disjoint paths from r may end at different vertices in a group [Chalermsook et al., SODA 2015], for which the authors gave a bicriteria approximation. For k3, there is no non-trivial approximation algorithm known for edge-weighted Restricted Group SNDP, except for the special case of the relaxed variant on trees (folklore). Our main result is an O(log n log h) approximation algorithm for Restricted Group SNDP that runs in time nf(k,w), where w is the treewidth of the input graph. Our algorithm works for both edge and vertex weighted variants, and the approximation ratio nearly matches the lower bound when k and w are constants. The key to achieving this result is a non-trivial extension of a framework introduced in [Chalermsook et al., SODA 2017]. This framework first embeds all feasible solutions to the problem into a dynamic program (DP) table. However, finding the optimal solution in the DP table remains intractable. We formulate a linear program relaxation for the DP and obtain an approximate solution via randomized rounding. This framework also allows us to systematically construct DP tables for high-connectivity problems. As a result, we present new exact algorithms for several variants of survivable network design problems in low-treewidth graphs.

Original language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018 |

Editors | Eric Blais, Jose D. P. Rolim, David Steurer, Klaus Jansen |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Print) | 9783959770859 |

DOIs | |

State | Published - 1 Aug 2018 |

Event | 21st International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2018 and the 22nd International Workshop on Randomization and Computation, RANDOM 2018 - Princeton, United States Duration: 20 Aug 2018 → 22 Aug 2018 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 116 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 21st International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2018 and the 22nd International Workshop on Randomization and Computation, RANDOM 2018 |
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Country/Territory | United States |

City | Princeton |

Period | 20/08/18 → 22/08/18 |