Polylogarithmic Approximation Algorithm for k-Connected Directed Steiner Tree on Quasi-Bipartite Graphs

Chun-Hsiang Chan; Bundit Laekhanukit; Hao-Ting Wei; Yuhao Zhang

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arXiv · 1911.09150

Polylogarithmic Approximation Algorithm for k-Connected Directed Steiner Tree on Quasi-Bipartite Graphs

Chun-Hsiang Chan·Bundit Laekhanukit·Hao-Ting Wei·Yuhao Zhang

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Abstract

In the k-Connected Directed Steiner Tree problem (k-DST), we are given a directed graph G=(V, E) with edge (or vertex) costs, a root vertex r, a set of q terminals T, and a connectivity requirement k>0; the goal is to find a minimum-cost subgraph H of G such that H has k internally disjoint paths from the root r to each terminal t . The k-DST problem is a natural generalization of the classical Directed Steiner Tree problem (DST) in the fault-tolerant setting in which the solution subgraph is required to have an r,t-path, for every terminal t, even after removing k-1 edges or vertices. Despite being a classical problem, there are not many positive results on the problem, especially for the case k >= 3. In this paper, we will present an O(log k log q)-approximation algorithm for k-DST when an input graph is quasi-bipartite, i.e., when there is no edge joining two non-terminal vertices. To the best of our knowledge, our algorithm is the only known non-trivial approximation algorithm for k-DST, for k >= 3, that runs in polynomial-time regardless of the structure of the optimal solution. In addition, our algorithm is tight for every constant k, due to the hardness result inherited from the Set Cover problem.

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Chun-Hsiang Chan, Bundit Laekhanukit, Hao-Ting Wei, Yuhao Zhang. 2019-11-20. Polylogarithmic Approximation Algorithm for k-Connected Directed Steiner Tree on Quasi-Bipartite Graphs. https://arxiv.org/abs/1911.09150

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