2 research outputs found
On Approximating Degree-Bounded Network Design Problems
Directed Steiner Tree (DST) is a central problem in combinatorial
optimization and theoretical computer science: Given a directed graph with edge costs , a root and
terminals , we need to output the minimum-cost arborescence in
that contains an \textrightarrow path for every . Recently,
Grandoni, Laekhanukit and Li, and independently Ghuge and Nagarajan, gave
quasi-polynomial time -approximation algorithms for the
problem, which are tight under popular complexity assumptions.
In this paper, we consider the more general Degree-Bounded Directed Steiner
Tree (DB-DST) problem, where we are additionally given a degree bound on
each vertex , and we require that every vertex in the output tree
has at most children. We give a quasi-polynomial time -bicriteria approximation: The algorithm produces a solution with
cost at most times the cost of the optimum solution that
violates the degree constraints by at most a factor of . This is
the first non-trivial result for the problem.
While our cost-guarantee is nearly optimal, the degree violation factor of
is an -factor away from the approximation lower bound
of from the set-cover hardness. The hardness result holds even
on the special case of the {\em Degree-Bounded Group Steiner Tree} problem on
trees (DB-GST-T). With the hope of closing the gap, we study the question of
whether the degree violation factor can be made tight for this special case. We
answer the question in the affirmative by giving an -bicriteria approximation algorithm for DB-GST-T
Quasi-polynomial Time Approximation Algorithm for Low-Degree Minimum-Cost Steiner Trees
In a recent paper [5], we addressed the problem of finding a minimum-cost spanning tree T for a given undirected graph G=(V,E) with maximum node-degree at most a given parameter B>1. We developed an algorithm based on Lagrangean relaxation that uses a repeated application of Kruskalrsquos MST algorithm interleaved with a combinatorial update of approximate Lagrangean node-multipliers maintained by the algorithm.
In this paper, we show how to extend this algorithm to the case of Steiner trees where we use a primal-dual approximation algorithm due to Agrawal, Klein, and Ravi [1] in place of Kruskalrsquos minimum-cost spanning tree algorithm. The algorithm computes a Steiner tree of maximum degree and total cost that is within a constant factor of that of a minimum-cost Steiner tree whose maximum degree is bounded by B. However, the running time is quasi-polynomial