3 research outputs found
Approximating Directed Steiner Problems via Tree Embedding
In the k-edge connected directed Steiner tree (k-DST) problem, we are given a
directed graph G on n vertices with edge-costs, a root vertex r, a set of h
terminals T and an integer k. The goal is to find a min-cost subgraph H of G
that connects r to each terminal t by k edge-disjoint r,t-paths. This problem
includes as special cases the well-known directed Steiner tree (DST) problem
(the case k = 1) and the group Steiner tree (GST) problem. Despite having been
studied and mentioned many times in literature, e.g., by Feldman et al.
[SODA'09, JCSS'12], by Cheriyan et al. [SODA'12, TALG'14] and by Laekhanukit
[SODA'14], there was no known non-trivial approximation algorithm for k-DST for
k >= 2 even in the special case that an input graph is directed acyclic and has
a constant number of layers. If an input graph is not acyclic, the complexity
status of k-DST is not known even for a very strict special case that k= 2 and
|T| = 2.
In this paper, we make a progress toward developing a non-trivial
approximation algorithm for k-DST. We present an O(D k^{D-1} log
n)-approximation algorithm for k-DST on directed acyclic graphs (DAGs) with D
layers, which can be extended to a special case of k-DST on "general graphs"
when an instance has a D-shallow optimal solution, i.e., there exist k
edge-disjoint r,t-paths, each of length at most D, for every terminal t. For
the case k= 1 (DST), our algorithm yields an approximation ratio of O(D log h),
thus implying an O(log^3 h)-approximation algorithm for DST that runs in
quasi-polynomial-time (due to the height-reduction of Zelikovsky
[Algorithmica'97]). Consequently, as our algorithm works for general graphs, we
obtain an O(D k^{D-1} log n)-approximation algorithm for a D-shallow instance
of the k-edge-connected directed Steiner subgraph problem, where we wish to
connect every pair of terminals by k-edge-disjoint paths
-Approximation Algorithm for Directed Steiner Tree: A Tight Quasi-Polynomial-Time Algorithm
In the Directed Steiner Tree (DST) problem we are given an -vertex
directed edge-weighted graph, a root , and a collection of terminal
nodes. Our goal is to find a minimum-cost arborescence that contains a directed
path from to every terminal. We present an -approximation algorithm for DST that runs in
quasi-polynomial-time. By adjusting the parameters in the hardness result of
Halperin and Krauthgamer, we show the matching lower bound of
for the class of quasi-polynomial-time
algorithms. This is the first improvement on the DST problem since the
classical quasi-polynomial-time approximation algorithm by
Charikar et al. (The paper erroneously claims an approximation due
to a mistake in prior work.)
Our approach is based on two main ingredients. First, we derive an
approximation preserving reduction to the Label-Consistent Subtree (LCST)
problem. The LCST instance has quasi-polynomial size and logarithmic height. We
remark that, in contrast, Zelikovsky's heigh-reduction theorem used in all
prior work on DST achieves a reduction to a tree instance of the related Group
Steiner Tree (GST) problem of similar height, however losing a logarithmic
factor in the approximation ratio. Our second ingredient is an LP-rounding
algorithm to approximately solve LCST instances, which is inspired by the
framework developed by Rothvo{\ss}. We consider a Sherali-Adams lifting of a
proper LP relaxation of LCST. Our rounding algorithm proceeds level by level
from the root to the leaves, rounding and conditioning each time on a proper
subset of label variables. A small enough (namely, polylogarithmic) number of
Sherali-Adams lifting levels is sufficient to condition up to the leaves
Polylogarithmic Approximation Algorithm for k-Connected Directed Steiner Tree on Quasi-Bipartite Graphs
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 edge-disjoint paths from the root r to each terminal in 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 vertices or edges. Despite being a classical problem, there are not many positive results on the problem, especially for the case k ? 3. In this paper, we 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 Our algorithm is tight for every constant k, due to the hardness result inherited from the Set Cover problem