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Optimal competitiveness for the Rectilinear Steiner Arborescence problem
We present optimal online algorithms for two related known problems involving
Steiner Arborescence, improving both the lower and the upper bounds. One of
them is the well studied continuous problem of the {\em Rectilinear Steiner
Arborescence} (). We improve the lower bound and the upper bound on the
competitive ratio for from and to
, where is the number of Steiner
points. This separates the competitive ratios of and the Symetric-,
two problems for which the bounds of Berman and Coulston is STOC 1997 were
identical. The second problem is one of the Multimedia Content Distribution
problems presented by Papadimitriou et al. in several papers and Charikar et
al. SODA 1998. It can be viewed as the discrete counterparts (or a network
counterpart) of . For this second problem we present tight bounds also in
terms of the network size, in addition to presenting tight bounds in terms of
the number of Steiner points (the latter are similar to those we derived for
)
A greedy approximation algorithm for the group Steiner problem
AbstractIn the group Steiner problem we are given an edge-weighted graph G=(V,E,w) and m subsets of vertices {gi}i=1m. Each subset gi is called a group and the vertices in ⋃igi are called terminals. It is required to find a minimum weight tree that contains at least one terminal from every group.We present a poly-logarithmic ratio approximation for this problem when the input graph is a tree. Our algorithm is a recursive greedy algorithm adapted from the greedy algorithm for the directed Steiner tree problem [Approximating the weight of shallow Steiner trees, Discrete Appl. Math. 93 (1999) 265–285, Approximation algorithms for directed Steiner problems, J. Algorithms 33 (1999) 73–91]. This is in contrast to earlier algorithms that are based on rounding a linear programming based relaxation for the problem [A polylogarithmic approximation algorithm for the Group Steiner tree problem, J. Algorithms 37 (2000) 66–84, preliminary version in Proceedings of SODA, 1998 pp. 253–259, On directed Steiner trees, Proceedings of SODA, 2002, pp. 59–63]. We answer in positive a question posed in [A polylogarithmic approximation algorithm for the Group Steiner tree problem, J. Algorithms 37 (2000) 66–84, preliminary version in Proceedings of SODA, 1998 pp. 253–259] on whether there exist good approximation algorithms for the group Steiner problem that are not based on rounding linear programs. For every fixed constant ε>0, our algorithm gives an O((log∑i|gi|)1+ε·logm) approximation in polynomial time. Approximation algorithms for trees can be extended to arbitrary undirected graphs by probabilistically approximating the graph by a tree. This results in an additional multiplicative factor of O(log|V|) in the approximation ratio, where |V| is the number of vertices in the graph. The approximation ratio of our algorithm on trees is slightly worse than the ratio of O(log(maxi|gi|)·logm) provided by the LP based approaches
Kruskal-Based Approximation Algorithm for the Multi-Level Steiner Tree Problem
We study the multi-level Steiner tree problem: a generalization of the
Steiner tree problem in graphs where terminals require varying priority,
level, or quality of service. In this problem, we seek to find a minimum cost
tree containing edges of varying rates such that any two terminals ,
with priorities , are connected using edges of rate
or better. The case where edge costs are proportional to
their rate is approximable to within a constant factor of the optimal solution.
For the more general case of non-proportional costs, this problem is hard to
approximate with ratio , where is the number of vertices in
the graph. A simple greedy algorithm by Charikar et al., however, provides a
-approximation in this setting, where
is an approximation ratio for a heuristic solver for the Steiner tree problem
and is the number of priorities or levels (Byrka et al. give a Steiner
tree algorithm with , for example).
In this paper, we describe a natural generalization to the multi-level case
of the classical (single-level) Steiner tree approximation algorithm based on
Kruskal's minimum spanning tree algorithm. We prove that this algorithm
achieves an approximation ratio at least as good as Charikar et al., and
experimentally performs better with respect to the optimum solution. We develop
an integer linear programming formulation to compute an exact solution for the
multi-level Steiner tree problem with non-proportional edge costs and use it to
evaluate the performance of our algorithm on both random graphs and multi-level
instances derived from SteinLib
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