Packing Element-Disjoint Steiner Trees. Mohammad R. Salavatipour Department of Computing Science University of Alberta Joint with Joseph Cheriyan Department of Combinatorics and Optimization University of Waterloo. The Problem. Given: Undirected graph G(V,E)
Mohammad R. Salavatipour
Department of Computing Science
University of Alberta
Department of Combinatorics and Optimization
University of Waterloo
We denote this problem by IUV
Observation: All leaves in a Steiner tree are terminals; otherwise, we can simply remove it.
Theorem (Menger): The maximum number of vertex-disjoint paths between two nod s,tis equal to the minimum number of vertices whose removal disconnect s,t
Theorem (Nash-Williams/Tutte): If the vertex-connectivity of Gis k then the maximum number of vertex-disjoint spanning trees in G is at least k/2
Theorem (Cheriyan & S.): It is NP-hard to approximate the maximum number of vertex-disjoint Steiner trees within a factor of (log n).
Packing edge-disjoint Steiner trees (IUE):
Conjecture (Kriesell’99): If the edge-connectivity of Gis k then the maximum number of edge-disjoint Steiner trees in Gis at least k/2
Theorem (Cheriyan & S.’04): The problem of packing element-disjoint directed Steiner trees is hard to approximate within (n1/3-). There is an O(n1/2+)-approximation for this problem.
A similar upper and lower bounds hold for packing edge-disjoint directed Steiner trees.
We settle down the approximability of IUV by giving an O(log n)-approximation algorithm even for a more general setting
Suppose we are given G(V,E), with terminals T, capacity cv for each vertex v2 V
Find a maximum size set of Steiner trees such that each vertex v is in at most cv trees
Let be the set of all Steiner trees in G
For every F2 let xF be a 0/1 variable
we can formulate the problem as an IP/LP
Main Theorem : There is a polytime randomized algorithm with ratio O(log n) for (uncapacitated) IUV. The algorithm finds a solution that is within a factor O(log n) of the optimal solution to the fractional IUV.
The same approximation ratio holds for capacitated IUV.
Since IUV (and even fractional IUV) is (log n)-hard we obtain:
Corollary: The approximability threshold of IUV is (log n).
We give the sketch of the proof for uncapacitated version
Let k be the largest vertex-connectivity between terminals
Clearly k is an upper bound for the (fractional) solution
The algorithm finds a set of element-disjoint Steiner trees.
First we reduce the problem to the bipartite case:
Bipartite IUV: if the input graph G is bipartite with one part being terminals and one part being Steiner points
Theorem: Given a graph G=(V,E) with terminal set T that is k-connected (and has no edge between terminals), there is a poly-time algorithm to obtain a bipartite graph G’ from G such that G’ has the same terminal set and is k-connected (on terminals).