Packing Element-Disjoint Steiner Trees

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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

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The Problem

• Given
• Undirected graph G(V,E)
• A set T of k terminals in G other vertices are
  called Steiner nodes

• Find
• Maximum number of Steiner trees that are disjoint
  on the Steiner nodes and the edges
  (element-disjoint)
• Steiner tree a tree that contains all the
  terminals T
• We denote this problem by IUV

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Example
Observation All leaves in a Steiner tree are
terminals otherwise, we can simply remove it.
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Special cases

• If the number of terminals is 2, k2 Steiner
  trees are paths
• So asking for maximum number of vertex-disjoint
  paths between two points
• Theorem (Menger) The maximum number of
  vertex-disjoint paths between two nod s,t is
  equal to the minimum number of vertices whose
  removal disconnect s,t

• The special case of TV Steiner trees are
  spanning trees
• Theorem (Nash-Williams/Tutte) If the
  vertex-connectivity of G is k then the maximum
  number of vertex-disjoint spanning trees in G is
  at least k/2

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From algorithmic point of view

• If k2 then the maximum number of vertex-disjoint
  paths can be easily found.

• Finding maximum number of vertex-disjoint
  spanning trees (i.e. when TV) can be solved in
  poly-time by matroid intersection algorithms.

• Finding maximum number of vertex-disjoint Steiner
  trees is NP-hard. In fact
• Theorem (Cheriyan S.) It is NP-hard to
  approximate the maximum number of vertex-disjoint
  Steiner trees within a factor of ?(log n).

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Other variations of the problem

• Packing edge-disjoint Steiner trees (IUE)
• The same input a graph G(V,E) and a set T of k
  terminals.
• The goal is to find maximum number of
  edge-disjoint Steiner trees.
• The special case of k2 is easy (by Mengers
  Theorem)
• The special case of TV Steiner trees are
  spanning trees again we can solve the problem in
  this case using matroid intersection algorithms

Conjecture (Kriesell99) If the
edge-connectivity of G is k then the maximum
number of edge-disjoint Steiner trees in G is at
least k/2
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Other variations of the problem (contd)

• Theorem (Lau04) If the edge-connectivity of G
  is k then the maximum number of edge-disjoint
  Steiner trees in G is at least k/26
• The known O(1)-approximations for IUE are based
  on solving a special case of IUV (bipartite)
• The problem is significantly harder on directed
  graphs
• 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.

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LP-formulation and generalization
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
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Fractional IUV

• The LP-relaxation of the problem will be
• The separation oracle for the dual of this LP is
  the minimum node-weighted Steiner tree problem
• Therefore, by a theorem of JMS03, even solving
  the LP is ?(log n)-hard

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Our Results

• 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

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Proof of the Main Theorem

• 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

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Proof of the Main Theorem (contd)

• By adding Steiner vertices on the edges we can
  assume there is no edge between terminals.
• So we have to pack Steiner trees that are
  disjoint on Steiner nodes only.
• Consider any edge euv.
• It can be shown that either deleting or
  contracting e preseves the connectivity of
  terminals. So
• 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).

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Proof of the Main Theorem (contd)

• Theorem 2 Given an instance of bipartite IUV
  such that the graph is k-connected, there is a
  randomized poly-time algorithm that finds a set
  of element-disjoint Steiner trees.
• Proof sketch
• Let and color Steiner nodes
  u.r. with one of k/R colors (c 6)
• Equivalently color every Steiner node u.r. with
  one of k colors 1,,k let Ci be color class i
• Partition the color classes into k/R
  super-classes where each super-class Dj consists
  of R consecutive color classes C(j-1)R1,
  C(j-1)R2,, CjR.
• Merge these classes one by one (in R rounds).

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Proof of the Main Theorem (contd)

• Let D1r be value of D1 after r rounds and HrD1r
  T
• Assume G1,,Gp are connected components of Hr
• It can be shown that when we add Cr1 to Hr ,
  the probability that G1 (or any fixed component
  of Hr) does not become connected to others is at
  most 1/e
• So, in expected, the number of connected
  components of Hr drops by a constant factor
  (1-1/e) in each round
• So after R rounds HR (which is DRT) becomes a
  connected graph, with probability at least 1-1/n
• Since there are k/R groups,