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

TL;DR

This paper introduces a polylogarithmic approximation algorithm for the k-Connected Directed Steiner Tree problem on quasi-bipartite graphs, providing the first polynomial-time solution for k >= 3 with proven tightness.

Contribution

It presents the first non-trivial polynomial-time approximation algorithm for k-DST on quasi-bipartite graphs for k >= 3, with a tight approximation ratio.

Findings

Achieves an O(log k log q)-approximation ratio.

Algorithm is the only known polynomial-time solution for k >= 3.

Approximation ratio is tight due to Set Cover hardness.

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.